The nonlinear fixed gravimetric boundary value problem
Institute of Scientific and Technical Information of China (English)
于锦海; 朱灼文
1995-01-01
The properly-posedness of the nonlinear fixed gravimetric boundary value problem is shown with the help of nonlinear functional analysis and a new iterative method to solve the problem is also given, where each step of the iterative program is reduced to solving one and the same kind of oblique derivative boundary value problem with the same type. Furthermore, the convergence of the iterative program is proved with Schauder estimate of elliptic differential equation.
Topological invariants in nonlinear boundary value problems
Energy Technology Data Exchange (ETDEWEB)
Vinagre, Sandra [Departamento de Matematica, Universidade de Evora, Rua Roma-tilde o Ramalho 59, 7000-671 Evora (Portugal)]. E-mail: smv@uevora.pt; Severino, Ricardo [Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga (Portugal)]. E-mail: ricardo@math.uminho.pt; Ramos, J. Sousa [Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisbon (Portugal)]. E-mail: sramos@math.ist.utl.pt
2005-07-01
We consider a class of boundary value problems for partial differential equations, whose solutions are, basically, characterized by the iteration of a nonlinear function. We apply methods of symbolic dynamics of discrete bimodal maps in the interval in order to give a topological characterization of its solutions.
THIRD-ORDER NONLINEAR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
王国灿; 金丽
2002-01-01
Third order singulary perturbed boundary value problem by means of differential inequality theories is studied. Based on the given results of second order nonlinear boundary value problem, the upper and lower solutions method of third order nonlinear boundary value problems by making use of Volterra type integral operator was established.Specific upper and lower solutions were constructed, and existence and asymptotic estimates of solutions under suitable conditions were obtained.The result shows that it seems to be new to apply these techniques to solving these kinds of third order singularly perturbed boundary value problem. An example is given to demonstrate the applications.
Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems
Directory of Open Access Journals (Sweden)
Boglaev Igor
2009-01-01
Full Text Available This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated, and convergence rates are estimated. Numerical experiments complement the theoretical results.
QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS WITH DISCONTINUOUS NONLINEARITIES
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
In this paper we shall consider a discontinuous nonlinear nonmonotone elliptic boundary value problem, i.e. a quasilinear elliptic hemivariational inequality. This kind of problems is strongly motivated by various problems in mechanics. By use of the notion of the generalized gradient of Clarke and the theory of pseudomonotone operators, we will prove the existence of solutions.
INITIAL BOUNDARY VALUE PROBLEM FOR A DAMPED NONLINEAR HYPERBOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
陈国旺
2003-01-01
In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equationare proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.
Nonlinear Second-Order Multivalued Boundary Value Problems
Indian Academy of Sciences (India)
Leszek Gasiński; Nikolaos S Papageorgiou
2003-08-01
In this paper we study nonlinear second-order differential inclusions involving the ordinary vector -Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operatory theory and from multivalued analysis, we obtain solutions for both the `convex' and `nonconvex' problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.
THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Mujeeb ur Rehman; Rahmat Ali Khan; Naseer Ahmad Asif
2011-01-01
In this paper,we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type cDδ0+u(t) =f(t,u(t),cDσ0+u(t)),t ∈[0,T],u(0) =αu(η),u(T) =βu(η),where1 ＜δ＜2,0＜σ＜ 1,α,β∈R,η∈(0,T),αη(1-β)+(1-α)(T-βη) ≠0 and cDoδ+,cDσ0+ are the Caputo fractional derivatives.We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results.Examples are also included to show the applicability of our results.
Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems
Directory of Open Access Journals (Sweden)
A. Boichuk
2011-01-01
Full Text Available Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems of n ordinary differential equations with constant coefficients and single delay (in the linear part and with a finite number of measurable delays of argument in nonlinearity: ż(t=Az(t-τ+g(t+εZ(z(hi(t,t,ε, t∈[a,b], assuming that these solutions satisfy the initial and boundary conditions z(s:=ψ(s if s∉[a,b], lz(⋅=α∈Rm. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional l does not coincide with the number of unknowns in the differential system with a single delay.
Institute of Scientific and Technical Information of China (English)
莫嘉琪
2003-01-01
The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.
Nonlinear boundary value problem for biregular functions in Clifford analysis
Institute of Scientific and Technical Information of China (English)
黄沙
1996-01-01
The biregular function in Clifford analysis is discussed. Plemelj’s formula is obtained andnonlinear boundary value problem: is considered. Applying the methodof integral equations and Schauder fixed-point theorem, the existence of solution for the above problem is proved.
A NONLOCAL NONLINEAR BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATIONS
Institute of Scientific and Technical Information of China (English)
YANJINHAI
1996-01-01
The existenoe and limit hehaviour of the solution for a kind of nonloeal noulinear boundary value condition on a part of the boundary is studied for the heat equation, which physicallymeans that the potential is the function of the total flux. When this part of boundary shrinks to a point in a certain way, this condition either results in a Dirac measure or simply disappears in the corresponding problem.
Institute of Scientific and Technical Information of China (English)
LiHongyu; SunJingxian
2005-01-01
By using topological method, we study a class of boundary value problem for a system of nonlinear ordinary differential equations. Under suitable conditions,we prove the existence of positive solution of the problem.
Existence of Solutions for Nonlinear Four-Point -Laplacian Boundary Value Problems on Time Scales
Directory of Open Access Journals (Sweden)
Topal SGulsan
2009-01-01
Full Text Available We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with a -Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result for -Laplacian boundary value problem is also given by the monotone method.
Institute of Scientific and Technical Information of China (English)
SU XIN-WEI
2011-01-01
This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects. The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in [B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(2009), 251258].
Energy Technology Data Exchange (ETDEWEB)
Zou, Li [Dalian Univ. of Technology, Dalian City (China). State Key Lab. of Structural Analysis for Industrial Equipment; Liang, Songxin; Li, Yawei [Dalian Univ. of Technology, Dalian City (China). School of Mathematical Sciences; Jeffrey, David J. [Univ. of Western Ontario, London (Canada). Dept. of Applied Mathematics
2017-06-01
Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.
Zou, Li; Liang, Songxin; Li, Yawei; Jeffrey, David J.
2017-03-01
Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.
Institute of Scientific and Technical Information of China (English)
鲁世平
2003-01-01
By employing the theory of differential inequality and some analysis methods, a nonlinear boundary value problem subject to a general kind of second-order Volterra functional differential equation was considered first. Then, by constructing the right-side layer function and the outer solution, a nonlinear boundary value problem subject to a kind of second- order Volterra functional differential equation with a small parameter was studied further. By using the differential mean value theorem and the technique of upper and lower solution, a new result on the existence of the solutions to the boundary value problem is obtained, and a uniformly valid asymptotic expansions of the solution is given as well.
Direct approach for solving nonlinear evolution and two-point boundary value problems
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2013-12-01
Time-delayed nonlinear evolution equations and boundary value problems have a wide range of applications in science and engineering. In this paper, we implement the differential transform method to solve the nonlinear delay differential equation and boundary value problems. Also, we present some numerical examples including time-delayed nonlinear Burgers equation to illustrate the validity and the great potential of the differential transform method. Numerical experiments demonstrate the use and computational efﬁciency of the method. This method can easily be applied to many nonlinear problems and is capable of reducing the size of computational work.
Initial-boundary value problems for a class of nonlinear thermoelastic plate equations
Institute of Scientific and Technical Information of China (English)
Zhang Jian-Wen; Rong Xiao-Liang; Wu Run-Heng
2009-01-01
This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions,it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations,by means of the Galerkin method. Moreover,it also proves the existence of strong and classical solutions.
Existence Theorems for Nonlinear Boundary Value Problems for Second Order Differential Inclusions
Kandilakis, Dimitrios A.; Papageorgiou, Nikolaos S.
1996-11-01
In this paper we consider a nonlinear two-point boundary value problem for second order differential inclusions. Using the Leray-Schauder principle and its multivalued analog due to Dugundji-Granas, we prove existence theorems for convex and nonconvex problems. Our results are quite general and incorporate as special cases several classes of problems which are of interest in the literature.
Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2-Dimensional Boundary Value Problems
Directory of Open Access Journals (Sweden)
Roman Cherniha
2015-08-01
Full Text Available A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k ≠ —2, for the power diffusivity uk when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k ≠ —2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity uk and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena to (1 + 1-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented.
A smart nonstandard finite difference scheme for second order nonlinear boundary value problems
Erdogan, Utku; Ozis, Turgut
2011-01-01
A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numer
A smart nonstandard finite difference scheme for second order nonlinear boundary value problems
Erdogan, Utku; Ozis, Turgut
2011-01-01
A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed.
Institute of Scientific and Technical Information of China (English)
Yepeng Xing; Qiong Wang; Valery G. Romanovski
2009-01-01
We prove several new comparison results and develop the monotone iterative tech-nique to show the existence of extremal solutions to a kind of periodic boundary value problem (PBVP) for nonlinear integro-differential equation of mixed type on time scales.
Positive Solutions of a Nonlinear Fourth-order Integral Boundary Value Problem
Directory of Open Access Journals (Sweden)
Benaicha Slimane
2016-07-01
Full Text Available In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.
THE NONLINEAR BOUNDARY VALUE PROBLEM FOR A CLASS OF INTEGRO-DIFFERENTIAL SYSTEM
Institute of Scientific and Technical Information of China (English)
Rongrong Tang
2006-01-01
In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.
EXISTENCE AND UNIQUENESS RESULTS FOR NONLINEAR THIRD-ORDER BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper,we investigate a nonlinear third-order three-point boundary value problem. By several well-known fixed point theorems,the existence of positive solutions is discussed. Besides,the uniqueness results are obtained by imposing growth restrictions on f.
Institute of Scientific and Technical Information of China (English)
Yaohong LI; Xiaoyan ZHANG
2013-01-01
In this paper,we consider boundary value problems for systems of nonlinear thirdorder differential equations.By applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem,the existence of multiple positive solutions is obtained.As application,we give some examples to demonstrate our results.
Existence of three solutions for impulsive nonlinear fractional boundary value problems
Directory of Open Access Journals (Sweden)
Shapour Heidarkhani
2017-01-01
Full Text Available In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.
Existence of Two Solutions of Nonlinear m-Point Boundary Value Problems
Institute of Scientific and Technical Information of China (English)
任景莉; 葛渭高
2003-01-01
Sufficient conditions for the existence of at least two positive solutions of a nonlinear m-points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.
Verified solutions of two-point boundary value problems for nonlinear oscillators
Bünger, Florian
Using techniques introduced by Nakao [4], Oishi [5, 6] and applied by Takayasu, Oishi, Kubo [11, 12] to certain nonlinear two-point boundary value problems (see also Rump [7], Chapter 15), we provide a numerical method for verifying the existence of weak solutions of two-point boundary value problems of the form -u″ = a(x, u) + b(x, u)u‧, 0 b are functions that fulfill some regularity properties. The numerical approximation is done by cubic spline interpolation. Finally, the method is applied to the Duffing, the van der Pol and the Toda oscillator. The rigorous numerical computations were done with INTLAB [8].
Roul, Pradip
2016-06-01
This paper presents a new iterative technique for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions. The method is based on the homotopy perturbation method and the integral equation formalism in which a recursive scheme is established for the components of the approximate series solution. This method does not involve solution of a sequence of nonlinear algebraic or transcendental equations for the unknown coefficients as in some other iterative techniques developed for singular boundary value problems. The convergence result for the proposed method is established in the paper. The method is illustrated by four numerical examples, two of which have physical significance: The first problem is an application of the reaction-diffusion process in a porous spherical catalyst and the second problem arises in the study of steady-state oxygen-diffusion in a spherical cell with Michaelis-Menten uptake kinetics.
Nonlinear boundary value problems for first order impulsive integro-differential equations
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1989-01-01
Full Text Available In this paper, we investigate a class of first order impulsive integro-differential equations subject to certain nonlinear boundary conditions and prove, with the help of upper and lower solutions, that the problem has a solution lying between the upper and lower solutions. We also develop monotone iterative technique and show the existence of multiple solutions of a class of periodic boundary value problems.
SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER
Institute of Scientific and Technical Information of China (English)
Wen Guochun
2007-01-01
The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.
EXISTENCE OF SOLUTIONS OF A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS IN L2-SPACES
Institute of Scientific and Technical Information of China (English)
WeiLi; ZhouHaiyun
2005-01-01
By using the perturbation results of sums of ranges of accretive mappings of Calvert and Gupta (1978),the abstract results on the existence of solutions of a family of nonlinear boundary value problems in L2 (Ω) are studied. The equation discussed in this paper and the methods used here are extension and complement to the corresponding results of Wei Li and He Zhen's previous papers. Especially,some new techniques are used in this paper.
Solvability of a three-point nonlinear boundary-value problem
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Assia Guezane-Lakoud
2010-09-01
Full Text Available Using the Leray Schauder nonlinear alternative, we prove the existence of a nontrivial solution for the three-point boundary-value problem $$displaylines{ u''+f(t,u= 0,quad 0
Positive solutions for a nonlinear periodic boundary-value problem with a parameter
Directory of Open Access Journals (Sweden)
Jingliang Qiu
2012-08-01
Full Text Available Using topological degree theory with a partially ordered structure of space, sufficient conditions for the existence and multiplicity of positive solutions for a second-order nonlinear periodic boundary-value problem are established. Inspired by ideas in Guo and Lakshmikantham [6], we study the dependence of positive periodic solutions as a parameter approaches infinity, $$ lim_{lambdao +infty}|x_{lambda}|=+infty,quadhbox{or}quad lim_{lambdao+infty}|x_{lambda}|=0. $$
Boundary Value Problems and Approximate Solutions
African Journals Online (AJOL)
Tadesse
boundary value problems suggested by nonlinear diffusion process. .... According to VIM, a correction functional could be written as follows: (5.4) ... The Variational Iteration Method is remarkably effective for solving boundary value problems.
The Modified Adomian Decomposition Method for Nonlinear Fractional Boundary Value Problems
Institute of Scientific and Technical Information of China (English)
WANG Jie
2012-01-01
We use the modified Adomian decomposition method(ADM) for solving the nonlinear fractional boundary value problem Dα0+u(x)=f(x,u(x)), 0＜x＜1, 3＜α≤4u(0) =α0, u″(0) =α2 (1)u(1) =β0, u″(1) =β2where Dα0+u is Caputo fractional derivative and α0,α2,β0,β2 is not zero at all,and f:[0,1] x R → R is continuous.The calculated numerical results show reliability and efficiency of the algorithm given.The numerical procedure is tested on linear and nonlinear problems.
Directory of Open Access Journals (Sweden)
Ying Wang
2015-03-01
Full Text Available In this article, we study the existence of multiple positive solutions for singular semipositone boundary-value problem (BVP with integral boundary conditions on infinite intervals. By using the properties of the Green's function and the Guo-Krasnosel'skii fixed point theorem, we obtain the existence of multiple positive solutions under conditions concerning the nonlinear functions. The method in this article can be used for a large number of problems. We illustrate the validity of our results with an example in the last section.
Nonlinear systems of differential inequalities and solvability of certain boundary value problems
Directory of Open Access Journals (Sweden)
Tvrdý Milan
2001-01-01
Full Text Available In the paper we present some new existence results for nonlinear second order generalized periodic boundary value problems of the form These results are based on the method of lower and upper functions defined as solutions of the system of differential inequalities associated with the problem and their relation to the Leray–Schauder topological degree of the corresponding operator. Our main goal consists in a fairly general definition of these functions as couples from . Some conditions ensuring their existence are indicated, as well.
Kounadis, A. N.
1992-05-01
An efficient and easily applicable, approximate analytic technique for the solution of nonlinear initial and boundary-value problems associated with nonlinear ordinary differential equations (O.D.E.) of any order and variable coefficients, is presented. Convergence, uniqueness and upper bound error estimates of solutions, obtained by the successive approximations scheme of the proposed technique, are thoroughly established. Important conclusions regarding the improvement of convergence for large time and large displacement solutions in case of nonlinear initial-value problems are also assessed. The proposed technique is much more efficient than the perturbations schemes for establishing the large postbuckling response of structural systems. The efficiency, simplicity and reliability of the proposed technique is demonstrated by two illustrative examples for which available numerical results exist.
Institute of Scientific and Technical Information of China (English)
SONG Li-mei; WENG Pei-xuan
2012-01-01
In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α ∈ (3,4],where the fractional derivative D0α+ is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.
A (k, n-k) Conjugate Boundary Value Problem with Semip ositone Nonlinearity
Institute of Scientific and Technical Information of China (English)
Yao Qing-liu; Shi Shao-yun
2015-01-01
The existence of positive solution is proved for a (k, n−k) conjugate boundary value problem in which the nonlinearity may make negative values and may be singular with respect to the time variable. The main results of Agarwal et al. (Agarwal R P, Grace S R, O’Regan D. Semipositive higher-order differential equa-tions. Appl. Math. Letters, 2004, 14: 201–207) are extended. The basic tools are the Hammerstein integral equation and the Krasnosel’skii’s cone expansion-compression technique.
Homotopy deform method for reproducing kernel space for nonlinear boundary value problems
Indian Academy of Sciences (India)
MIN-QIANG XU; YING-ZHEN LIN
2016-10-01
In this paper, the combination of homotopy deform method (HDM) and simplified reproducing kernel method (SRKM) is introduced for solving the boundary value problems (BVPs) of nonlinear differential equations. The solution methodology is based on Adomian decomposition and reproducing kernel method (RKM). By the HDM, the nonlinear equations can be converted into a series of linear BVPs. After that, the simplified reproducing kernel method, which not only facilitates the reproducing kernel but also avoids the time-consuming Schmidt orthogonalization process, is proposed to solve linear equations. Some numerical test problems including ordinary differential equations and partial differential equations are analysed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that our algorithm is considerably accurate and effective as expected.
Mawhin, Jean; Ure??a, Antonio J.
2002-01-01
A generalization of the well-known Hartman-Nagumo inequality to the case of the vector ordinary p-Laplacian and classical degree theory provide existence results for some associated nonlinear boundary value problems.
Directory of Open Access Journals (Sweden)
Ureña Antonio J
2002-01-01
Full Text Available A generalization of the well-known Hartman–Nagumo inequality to the case of the vector ordinary -Laplacian and classical degree theory provide existence results for some associated nonlinear boundary value problems.
Institute of Scientific and Technical Information of China (English)
高永馨
2002-01-01
Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equation y(4n)= f( t,y,y' ,y",… ,y(4n－1) ) (a) with the boundary conditions g2i(y(2i) (a) ,y(2i+1) (a)) = 0,h2i(y(2i) (c) ,y(2i+1) (c)) = 0, (I= 0,1,…,2n － 1 ) (b) where the functions f, gi and hi are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equation y(n) = f(t,y,y',y",… ,y(n－1)) many results have been given at the present time. But the existence of solutions of boundary value problem (a), (b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, I.e. Existence of solutions of the boundary value problem. Y(4n) = f(t,y,y',y",… ,y(4n－1) ) a2iy(2i) (at) + a2i+1y(2i+1) (a) = b2i ,c2iy(2O ( c ) + c2i+1y(2i+1) ( c ) = d2i, ( I = 0,1 ,…2n － 1) has not been dealt with in previous works.
Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems
Energy Technology Data Exchange (ETDEWEB)
Massoudi, M.C.; Tran, P.X.
2006-01-01
We discuss three boundary value problems in the flow and heat transfer analysis in flowing granular materials: (i) the flow down an inclined plane with radiation effects at the free surface; (ii) the natural convection flow between two heated vertical walls; (iii) the shearing motion between two horizontal flat plates with heat conduction. It is assumed that the material behaves like a continuum, similar to a compressible nonlinear fluid where the effects of density gradients are incorporated in the stress tensor. For a fully developed flow the equations are simplified to a system of three nonlinear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, and so forth are presented.
Renormalization-group symmetries for solutions of nonlinear boundary value problems
Kovalev, V F
2008-01-01
Approximately 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov renormalization group treated as a Lie group of continuous transformations. Overwhelmingly dominating practical quantum field theory calculations, the renormalization-group method formed the basis for the discovery of the asymptotic freedom of strong nuclear interactions and underlies the Grand Unification scenario. This paper describes the logical framework of a new algorithm based on the modern theory of transformation groups and presents the most interesting results of application of the method to differential and/or integral equation problems and to problems that involve linear functionals of solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics are given, where new analytical solutions obtained with this algorithm have allowed describing the singular...
Institute of Scientific and Technical Information of China (English)
Jia-qi Mo; Wan-tao Lin
2006-01-01
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.
Institute of Scientific and Technical Information of China (English)
Liu YANG; Zongmin QIAO
2012-01-01
In this paper,the existence and multiplicity of positive solutions for Robin type boundary value problem of differential equation involving the Riemann-Liouville fractional order derivative are established.
An efficient numerical technique for the solution of nonlinear singular boundary value problems
Singh, Randhir; Kumar, Jitendra
2014-04-01
In this work, a new technique based on Green's function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green's function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.
Institute of Scientific and Technical Information of China (English)
李仁贵; 刘立山
2001-01-01
New existence results are presented for the singular second-order nonlinear boundary value problems u" + g(t)f(u) = 0, 0 ＜ t ＜ 1, au(0) - βu′(0) = 0,γu(1) +δu'(l) = 0 under the conditions 0 ≤ fn+ ＜ M1, m1 ＜ f∞-≤∞ or 0 ≤ f∞+＜M1, m1 ＜ f 0-≤ ∞, where f +0＝ limu→of(u)/u, f∞-＝ limu-→∞(u)/u, f0-＝limu-→of(u)/u, f+∞＝ limu→=f(u)/u, g may be singular att ＝ 0 and/ort ＝ 1 . Theproof uses a fixed point theorem in cone theory.
Modelling of hydrogen thermal desorption spectrum in nonlinear dynamical boundary-value problem
Kostikova, E. K.; Zaika, Yu V.
2016-11-01
One of the technological challenges for hydrogen materials science (including the ITER project) is the currently active search for structural materials with various potential applications that will have predetermined limits of hydrogen permeability. One of the experimental methods is thermal desorption spectrometry (TDS). A hydrogen-saturated sample is degassed under vacuum and monotone heating. The desorption flux is measured by mass spectrometer to determine the character of interactions of hydrogen isotopes with the solid. We are interested in such transfer parameters as the coefficients of diffusion, dissolution, desorption. The paper presents a distributed boundary-value problem of thermal desorption and a numerical method for TDS spectrum simulation, where only integration of a nonlinear system of low order (compared with, e.g., the method of lines) ordinary differential equations (ODE) is required. This work is supported by the Russian Foundation for Basic Research (project 15-01-00744).
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
The iterative technique of sign-changing solution is studied for a nonlinear third-order two-point boundary value problem, where the nonlinear term has the time sin-gularity. By applying the monotonically iterative technique, an existence theorem is established and two useful iterative schemes are obtained.
Initial-Boundary Value Problem Solution of the Nonlinear Shallow-water Wave Equations
Kanoglu, U.; Aydin, B.
2014-12-01
The hodograph transformation solutions of the one-dimensional nonlinear shallow-water wave (NSW) equations are usually obtained through integral transform techniques such as Fourier-Bessel transforms. However, the original formulation of Carrier and Greenspan (1958 J Fluid Mech) and its variant Carrier et al. (2003 J Fluid Mech) involve evaluation integrals. Since elliptic integrals are highly singular as discussed in Carrier et al. (2003), this solution methodology requires either approximation of the associated integrands by smooth functions or selection of regular initial/boundary data. It should be noted that Kanoglu (2004 J Fluid Mech) partly resolves this issue by simplifying the resulting integrals in closed form. Here, the hodograph transform approach is coupled with the classical eigenfunction expansion method rather than integral transform techniques and a new analytical model for nonlinear long wave propagation over a plane beach is derived. This approach is based on the solution methodology used in Aydın & Kanoglu (2007 CMES-Comp Model Eng) for wind set-down relaxation problem. In contrast to classical initial- or boundary-value problem solutions, here, the NSW equations are formulated to yield an initial-boundary value problem (IBVP) solution. In general, initial wave profile with nonzero initial velocity distribution is assumed and the flow variables are given in the form of Fourier-Bessel series. The results reveal that the developed method allows accurate estimation of the spatial and temporal variation of the flow quantities, i.e., free-surface height and depth-averaged velocity, with much less computational effort compared to the integral transform techniques such as Carrier et al. (2003), Kanoglu (2004), Tinti & Tonini (2005 J Fluid Mech), and Kanoglu & Synolakis (2006 Phys Rev Lett). Acknowledgments: This work is funded by project ASTARTE- Assessment, STrategy And Risk Reduction for Tsunamis in Europe. Grant 603839, 7th FP (ENV.2013.6.4-3 ENV
EXISTENCE OF SOLUTIONS TO A CLASS OF NONLINEAR n-DIMENSIONAL DISCRETE BOUNDARY VALUE PROBLEMS
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无
2012-01-01
In this paper,using the critical point theory,we obtain a new result on the existence of the solutions to a class of n-dimensional discrete boundary value problems.Results obtained extend or improve the existing ones.
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Shanying Zhu
2009-01-01
This paper deals with the existence of positive solutions to the singular second-order periodic boundary value problem, We obtain the existence results of positive solutions by the fixed point index theory. The results obtained extend and complement some known results.
奇摄动非线性边值问题%THE SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEMS
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莫嘉琪
2000-01-01
The singularly perturbed nonlinear boundary value problems are considered.Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained.And then the uniform validity of solution is proved by using the differential inequalities.
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Peiguo Zhang
2013-01-01
Full Text Available By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.
Peter E. Zhidkov
2001-01-01
We find sufficient conditions for systems of functions to be Riesz bases in $L_2(0,1)$. Then we improve a theorem presented in [13] by showing that a ``standard'' system of solutions of a nonlinear boundary-value problem, normalized to 1, is a Riesz basis in $L_2(0,1)$. The proofs in this article use Bari's theorem.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we consider a singular nth order three-point boundary value problem with sign changing nonlinearity. By the method of lower solution and topology degree theorem, we investigate the existence of positive solutions to the above problem. Moreover, the associated Green’s function for the above problem is also given. The results of this paper are new and extend the previous known results.
Boundary Value Problems With Integral Conditions
Karandzhulov, L. I.; Sirakova, N. D.
2011-12-01
The weakly perturbed nonlinear boundary value problems (BVP) for almost linear systems of ordinary differential equations (ODE) are considered. We assume that the nonlinear part contain an additional function, which defines the perturbation as singular. Then the Poincare method is not applicable. The problem of existence, uniqueness and construction of a solution of the posed BVP with integral condition is studied.
Fixed set theorems for discrete dynamics and nonlinear boundary-value problems
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Robert Brooks
2011-05-01
Full Text Available We consider self-mappings of Hausdorff topological spaces which map compact sets to compact sets and establish the existence of invariant (fixed sets. The fixed set results are used to provide fixed set analogues of well-known fixed point theorems. An algorithm is employed to compute the existence of fixed sets which are self-similar in a generalized sense. Some numerical examples are given. The utility of the abstract result is further illustrated via the study of a boundary value problem for a system of differential equations
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WANG Rouhuai
2006-01-01
The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems.It is proved that if the corresponding first variation is regular in Lopatinski(i) sense,then the solution is analytic up to the boundary.The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich,and hence completely generalize the previous result of C.B.Morrey.The author also discusses linear elliptic boundary value problems for systems of ellip tic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients.Combining the standard Fourier transform technique with analytic continuation argument,the author constructs the Poisson and Green's kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions.Some a priori estimates of Schauder type and Lp type are obtained.
Energy Technology Data Exchange (ETDEWEB)
Kong Dexing [Department of Mathematics, Zhejiang University, Hangzhou 310027 (China); Sun Qingyou, E-mail: qysun@cms.zju.edu.cn [Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027 (China)
2011-04-01
All articles must In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.
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Peter E. Zhidkov
2001-12-01
Full Text Available We find sufficient conditions for systems of functions to be Riesz bases in $L_2(0,1$. Then we improve a theorem presented in [13] by showing that a ``standard'' system of solutions of a nonlinear boundary-value problem, normalized to 1, is a Riesz basis in $L_2(0,1$. The proofs in this article use Bari's theorem.
Liang Yue; Yang He
2011-01-01
Abstract The paper deals with the existence of positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations - u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = θ and u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) ...
Solvability of Third-order Three-point Boundary Value Problems with Carathéodory Nonlinearity
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YAO QING-LIU; Shi Shao-yun
2012-01-01
A class of third-order three-point boundary value problems is considered,where the nonlinear term is a Carathéodory function.By introducing a height function and considering the integration of this height function,an existence theorem of solution is proved when the limit growth function exists.The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.
Non-normal Hasemann Boundary Value Problem
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
We will discuss the non-normal Hasemann boundary value problem:we may find these results are coincided with those of normal Hasemann boundary value problem and non normal Riemann boundary value problem.
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WEI Li; ZHOU Haiyun
2005-01-01
By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we study the abstract results on the existence of a solution u ∈ Ls (Ω) of nonlinear boundary value problems involving the p-Laplacian operator, where2 ≤ s ＜ +∞, and 2N/N+1 ＜ p ≤ 2 for N(≥ 1) which denotes the dimension of RN. To obtain the result, some new techniques are used in this paper. The equation discussed in this paper and our methods here are extension and complement to the corresponding results of L. Wei and Z. He.
Multipoint Singular Boundary-Value Problem for Systems of Nonlinear Differential Equations
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Zdeněk Šmarda
2009-01-01
Full Text Available A singular Cauchy-Nicoletti problem for a system of nonlinear ordinary differential equations is considered. With the aid of combination of Ważewski's topological method and Schauder's principle, the theorem concerning the existence of a solution of this problem (having the graph in a prescribed domain is proved.
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Cristian Enache
2006-06-01
Full Text Available For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of u(x and |Ã¢ÂˆÂ‡u|2, where u(x are the solutions of our problems. From these inequalities, we derive, using Hopf's maximum principles, some maximum principles for the appropriate combinations of u(x and |Ã¢ÂˆÂ‡u|2, and we list a few examples of problems to which these maximum principles may be applied.
Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions
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Nemat Nyamoradi
2013-01-01
Full Text Available We consider a system of boundary value problems for fractional differential equation given by D0+βϕp(D0+αu(t=λ1a1(tf1(u(t,v(t, t∈(0,1, D0+βϕp(D0+αv(t=λ2a2(tf2(u(t,v(t, t∈(0,1, where 1<α, β≤2, 2<α+β≤4, λ1, λ2 are eigenvalues, subject either to the boundary conditions D0+αu(0=D0+αu(1=0, u(0=0, D0+β1u(1-Σi=1m-2a1i D0+β1u(ξ1i=0, D0+αv(0=D0+αv(1=0, v(0=0, D0+β1v(1-Σi=1m-2a2i D0+β1v(ξ2i=0 or D0+αu(0=D0+αu(1=0, u(0=0, D0+β1u(1-Σi=1m-2a1i D0+β1u(ξ1i=ψ1(u, D0+αv(0=D0+αv(1=0, v(0=0, D0+β1v(1-Σi=1m-2a2i D0+β1v(ξ2i=ψ2(v, where 0<β1<1, α-β1-1≥0 and ψ1, ψ2:C([0,1]→[0, ∞ are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.
Zhai, Chengbo; Hao, Mengru
2014-01-01
By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by -D(0+)(ν1)y1(t) = λ1a1(t)f(y1(t), y2(t)), - D(0+)(ν2)y2(t) = λ2a2(t)g(y1(t), y2(t)), where D(0+)(ν) is the standard Riemann-Liouville fractional derivative, ν1, ν2 ∈ (n - 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = 0 = [D(0+ (α)y2(t)] t=1, for 1 ≤ α ≤ n - 2, or y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = ϕ1(y1), [D(0+)(α)y2(t)] t=1 = ϕ2(y2), for 1 ≤ α ≤ n - 2, ϕ1, ϕ2 ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.
Timergaliev, S. N.
2009-06-01
This paper deals with the proof of the existence of solutions of a geometrically and physically nonlinear boundary value problem for shallow Timoshenko shells with the transverse shear strains taken into account. The shell edge is assumed to be partly fixed. It is proposed to study the problem by a variational method based on searching the points of minimum of the total energy functional for the shell-load system in the space of generalized displacements. We show that there exists a generalized solution of the problemon which the total energy functional attains its minimum on a weakly closed subset of the space of generalized displacements.
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Cheng Xiaoliang; Ying Weiting
2005-01-01
In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
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Said Mesloub
2008-03-01
Full Text Available This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
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Mesloub Said
2008-01-01
Full Text Available This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.
Payette, G. S.; Reddy, J. N.
2011-05-01
In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L2 norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier-Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.
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Liaqat Ali
2016-09-01
Full Text Available In this research work a new version of Optimal Homotopy Asymptotic Method is applied to solve nonlinear boundary value problems (BVPs in finite and infinite intervals. It comprises of initial guess, auxiliary functions (containing unknown convergence controlling parameters and a homotopy. The said method is applied to solve nonlinear Riccati equations and nonlinear BVP of order two for thin film flow of a third grade fluid on a moving belt. It is also used to solve nonlinear BVP of order three achieved by Mostafa et al. for Hydro-magnetic boundary layer and micro-polar fluid flow over a stretching surface embedded in a non-Darcian porous medium with radiation. The obtained results are compared with the existing results of Runge-Kutta (RK-4 and Optimal Homotopy Asymptotic Method (OHAM-1. The outcomes achieved by this method are in excellent concurrence with the exact solution and hence it is proved that this method is easy and effective.
An iterative HAM approach for nonlinear boundary value problems in a semi-infinite domain
Zhao, Yinlong; Lin, Zhiliang; Liao, Shijun
2013-09-01
In this paper, we propose an iterative approach to increase the computation efficiency of the homotopy analysis method (HAM), a analytic technique for highly nonlinear problems. By means of the Schmidt-Gram process (Arfken et al., 1985) [15], we approximate the right-hand side terms of high-order linear sub-equations by a finite set of orthonormal bases. Based on this truncation technique, we introduce the Mth-order iterative HAM by using each Mth-order approximation as a new initial guess. It is found that the iterative HAM is much more efficient than the standard HAM without truncation, as illustrated by three nonlinear differential equations defined in an infinite domain as examples. This work might greatly improve the computational efficiency of the HAM and also the Mathematica package BVPh for nonlinear BVPs.
Positive Solutions for Nonlinear nth-Order Singular Nonlocal Boundary Value Problems
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Xin'an Hao
2007-04-01
Full Text Available We study the existence and multiplicity of positive solutions for a class of nth-order singular nonlocal boundary value problemsu(n(t+a(tf(t,u=0, tÃ¢ÂˆÂˆ(0,1, u(0=0, u'(0=0, Ã¢Â€Â¦,u(nÃ¢ÂˆÂ’2(0=0, ÃŽÂ±u(ÃŽÂ·=u(1, where 0<ÃŽÂ·<1,Ã¢Â€Â‰Ã¢Â€Â‰0<ÃŽÂ±ÃŽÂ·nÃ¢ÂˆÂ’1Ã¢Â€Â‰<1. The singularity may appear at t=0 and/or t=1. The Krasnosel'skii-Guo theorem on cone expansion and compression is used in this study. The main results improve and generalize the existing results.
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I. J. Cabrera
2012-01-01
Full Text Available We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: D0+αu(t+f(t,u(t=0, 0
Caixia Guo; Jianmin Guo; Ying Gao; Shugui Kang
2016-01-01
This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established.
Group invariance in engineering boundary value problems
Seshadri, R
1985-01-01
REFEREN CES . 156 9 Transforma.tion of a Boundary Value Problem to an Initial Value Problem . 157 9.0 Introduction . 157 9.1 Blasius Equation in Boundary Layer Flow . 157 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods . 163 9.3 Summary . 168 REFERENCES . . . . . . . . . . . . . . . . . . 168 . 10 From Nonlinear to Linear Differential Equa.tions Using Transformation Groups. . . . . . . . . . . . . . 169 . 10.1 From Nonlinear to Linear Differential Equations . 170 10.2 Application to Ordinary Differential Equations -Bernoulli's Equation . . . . . . . . . . . 173 10.3 Application to Partial Differential Equations -A Nonlinear Chemical Exchange Process . 178 10.4 Limitations of the Inspectional Group Method . 187 10.5 Summary . 188 REFERENCES . . . . 188 11 Miscellaneous Topics . 190 11.1 Reduction of Differential Equations to Algebraic Equations 190 11.2 Reduction of Order of an Ordinary Differential Equation . 191 11.3 Transformat.ion From Ordinary to Partial Differential Equations-Search for First Inte...
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Lv Xuezhe
2010-01-01
Full Text Available Abstract The existence and uniqueness of positive solution is obtained for the singular second-order -point boundary value problem for , , , where , , are constants, and can have singularities for and/or and for . The main tool is the perturbation technique and Schauder fixed point theorem.
Zhidkov, P E
2001-01-01
We establish examples of systems of functions being Riesz bases in L_{2}(0,1). We then apply this result to improve a theorem presented in [9] showing that an arbitrary "standard" system of solutions of a nonlinear boundary value problem, normalized to 1 in the same space, is a Riesz basis in this space. The proofs in this work are quite elementary.
Boundary value problems and partial differential equations
Powers, David L
2005-01-01
Boundary Value Problems is the leading text on boundary value problems and Fourier series. The author, David Powers, (Clarkson) has written a thorough, theoretical overview of solving boundary value problems involving partial differential equations by the methods of separation of variables. Professors and students agree that the author is a master at creating linear problems that adroitly illustrate the techniques of separation of variables used to solve science and engineering.* CD with animations and graphics of solutions, additional exercises and chapter review questions* Nearly 900 exercises ranging in difficulty* Many fully worked examples
BOUNDARY VALUE PROBLEM TO DYNAMIC EQUATION ON TIME SCALE
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无
2011-01-01
In this paper we consider a nonlinear first-order boundary value problem on a time scale. The existence results of three positive solutions are obtained using fixed point theorems. Finally,examples are presented to illustrate the main results.
Boundary value problems for partial differential equations with exponential dichotomies
Laederich, Stephane
We are extending the notion of exponential dichotomies to partial differential evolution equations on the n-torus. This allows us to give some simple geometric criteria for the existence of solutions to certain nonlinear Dirichlet boundary value problems.
Riemann Boundary Value Problems for Koch Curve
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Zhengshun Ruanand
2012-11-01
Full Text Available In this study, when L is substituted for Koch curve, Riemann boundary value problems was defined, but generally speaking, Cauchy-type integral is meaningless on Koch curve. When some analytic conditions are attached to functions G (z and g (z, through the limit function of a sequence of Cauchytype integrals, the homogeneous and non-homogeneous Riemann boundary problems on Koch curve are introduced, some similar results was attained like the classical boundary value problems for analytic functions.
Boundary value problems and Markov processes
Taira, Kazuaki
1991-01-01
Focussing on the interrelations of the subjects of Markov processes, analytic semigroups and elliptic boundary value problems, this monograph provides a careful and accessible exposition of functional methods in stochastic analysis. The author studies a class of boundary value problems for second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems, and proves that this class of boundary value problems provides a new example of analytic semigroups both in the Lp topology and in the topology of uniform convergence. As an application, one can construct analytic semigroups corresponding to the diffusion phenomenon of a Markovian particle moving continuously in the state space until it "dies", at which time it reaches the set where the absorption phenomenon occurs. A class of initial-boundary value problems for semilinear parabolic differential equations is also considered. This monograph will appeal to both advanced students and researchers as an introductio...
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Baoqiang Yan
2008-10-01
Full Text Available Using the fixed point theorem in cones, this paper shows the existence of multiple positive solutions for the singular $m$-point boundary-value problem $$displaylines{ x''(t+a(tf(t,x(t,x'(t=0,quad 0
Fourier analysis and boundary value problems
Gonzalez-Velasco, Enrique A
1996-01-01
Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics.A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.Key Features* Topics are covered from a historical perspective with biographical information on key contributors to the field* The text contains more than 500 exercises* Includes practical applicati...
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无
2001-01-01
In this paper,Fucik spectrum,ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and some new results on the existence of nontrivial solutions,multiple solutions and sign-changing solutions are obtained.In one case seven nontrivial solutions are got.The techniques have independent interest.
Fast Integration of One-Dimensional Boundary Value Problems
Campos, Rafael G.; Ruiz, Rafael García
2013-11-01
Two-point nonlinear boundary value problems (BVPs) in both unbounded and bounded domains are solved in this paper using fast numerical antiderivatives and derivatives of functions of L2(-∞, ∞). This differintegral scheme uses a new algorithm to compute the Fourier transform. As examples we solve a fourth-order two-point boundary value problem (BVP) and compute the shape of the soliton solutions of a one-dimensional generalized Korteweg-de Vries (KdV) equation.
Semigroups, boundary value problems and Markov processes
Taira, Kazuaki
2014-01-01
A careful and accessible exposition of functional analytic methods in stochastic analysis is provided in this book. It focuses on the interrelationship between three subjects in analysis: Markov processes, semi groups and elliptic boundary value problems. The author studies a general class of elliptic boundary value problems for second-order, Waldenfels integro-differential operators in partial differential equations and proves that this class of elliptic boundary value problems provides a general class of Feller semigroups in functional analysis. As an application, the author constructs a general class of Markov processes in probability in which a Markovian particle moves both by jumps and continuously in the state space until it 'dies' at the time when it reaches the set where the particle is definitely absorbed. Augmenting the 1st edition published in 2004, this edition includes four new chapters and eight re-worked and expanded chapters. It is amply illustrated and all chapters are rounded off with Notes ...
Riemann boundary value problem for hyperanalytic functions
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Ricardo Abreu Blaya
2005-01-01
Full Text Available We deal with Riemann boundary value problem for hyperanalytic functions. Furthermore, necessary and sufficient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.
POSITIVE SOLUTION FOR NONLINEARSINGULAR BOUNDARY VALUE PROBLEMS
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吕海深
2004-01-01
Under suitable conditions on f(·, u), it is shown that the two-point boundary value problem(φp(u'))'+ λq(t)f(u)=0 in (0, 1),u(0)=u(1)=0,has two positive solution or at least one positive solution for A in a compatible interval.
Initial boundary value problems in mathematical physics
Leis, Rolf
2013-01-01
Based on the author's lectures at the University of Bonn in 1983-84, this book introduces classical scattering theory and the time-dependent theory of linear equations in mathematical physics. Topics include proof of the existence of wave operators, some special equations of mathematical physics, exterior boundary value problems, radiation conditions, and limiting absorption principles. 1986 edition.
Separable boundary-value problems in physics
Willatzen, Morten
2011-01-01
Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level. Separable Boundary-Value Problems in Physics is an accessible and comprehensive treatment of partial differential equations i
Homology in Electromagnetic Boundary Value Problems
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Matti Pellikka
2010-01-01
Full Text Available We discuss how homology computation can be exploited in computational electromagnetism. We represent various cellular mesh reduction techniques, which enable the computation of generators of homology spaces in an acceptable time. Furthermore, we show how the generators can be used for setting up and analysis of an electromagnetic boundary value problem. The aim is to provide a rationale for homology computation in electromagnetic modeling software.
Superlinear singular fractional boundary-value problems
Directory of Open Access Journals (Sweden)
Imed Bachar
2016-04-01
Full Text Available In this article, we study the superlinear fractional boundary-value problem $$\\displaylines{ D^{\\alpha }u(x =u(xg(x,u(x,\\quad 00$. The function $g(x,u\\in C((0,1\\times [ 0,\\infty ,[0,\\infty$ that may be singular at x=0 and x=1 is required to satisfy convenient hypotheses to be stated later.
Directory of Open Access Journals (Sweden)
Chi-Chang Wang
2013-09-01
Full Text Available This paper seeks to use the proposed residual correction method in coordination with the monotone iterative technique to obtain upper and lower approximate solutions of singularly perturbed non-linear boundary value problems. First, the monotonicity of a non-linear differential equation is reinforced using the monotone iterative technique, then the cubic-spline method is applied to discretize and convert the differential equation into the mathematical programming problems of an inequation, and finally based on the residual correction concept, complex constraint solution problems are transformed into simpler questions of equational iteration. As verified by the four examples given in this paper, the method proposed hereof can be utilized to fast obtain the upper and lower solutions of questions of this kind, and to easily identify the error range between mean approximate solutions and exact solutions.
POSITIVE SOLUTIONS TO FOURTH-ORDER NEUMANN BOUNDARY VALUE PROBLEM
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无
2010-01-01
In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.
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LI; Shujie
2001-01-01
［1］Martin Schecher,The Fucik spectrum,Indiana University Mathematics Journal,1994,43(4):1139-1157.［2］Dancer,E.N.,Remarks on jumping nonlinearities,in Topics in Nonlinear Analysis (eds.Escher,Simonett),Basel:Birkhauser,1999,101-116.［3］Dancer,E.N.,Du Yihong,Existence of changing sign solutions for semilinear problems with jumping nonlinearities at zero,Proceedings of the Royal Society of Edinburgh,1994,124A:1165-1176.［4］Dancer,E.N.,Du Yihong,Multiple solutions of some semilinear elliptic equations via generalized conley index,Journal of Mathematical Analysis and Applications,1995,189:848-871.［5］Li Shujie,Zhang Zhitao,Sign-changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities,Acta Mathematica Sinica,2000,16(1):113-122.［6］Chang Kung-ching,Li Shujie,Liu Jiaquan,Remarks on multiple solutions for asymptotically linear elliptic boundary value problems.Topological methods for Nonlinear Analysis,Journal of the Juliusz Schauder Center,1994,3:179-187.［7］Alfonso Castro,Jorge Cossio,Multiple solutions for a nonlinear Dirichlet problem,SIAM J.Math.Anal.,1994,25(6):1554-1561.［8］Alfonso Castro,Jorge Cossio,John M.Neuberger,A sign-changing solution for a superlinear Dirichlet problem,Rocky Mountain J.M.,1997,27:1041-1053.［9］Alfonso Castro,Jorge Cossio,John M.Neuberger,A minimax principle,index of the critical point,and existence of sign-changing solutions to Elliptic boundary value problems,E.J.Diff.Eqn.,1998 (2):1-18.［10］Thomas Bartsch,Wang Zhiqiang,On the existence of sign-changing solutions for semilinear Dirichlet problems,Topological Methods in Nonlinear Analysis,Journal of the Juliusz Schauder Center,1996,(7):115-131.［11］Li Shujie,Zhang Zhitao,Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity,Discrete and Continuous Dynamical System,1999,5(3):489-493.［12］Mawhin,J.,Willem,M.,Critical Point Theory and
Institute of Scientific and Technical Information of China (English)
李志龙
2008-01-01
In this paper, we study the nonlinear second-order boundary value problem of delay differential equation. Without the assumption of the nonnegativity of f, we still obtain the existence of the positive solution.
Institute of Scientific and Technical Information of China (English)
杨茂; 陈建军
1999-01-01
In this paper,combining Riemann's method with the fixed point theory effectively,we proved that the migration equation of the moisture in soil with nonlinear initial boundary value problem has unique classical solution.
Complementary Lidstone Interpolation and Boundary Value Problems
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Pinelas Sandra
2009-01-01
Full Text Available We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial of degree , which involves interpolating data at the odd-order derivatives. For we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a th order differential equation and the complementary Lidstone boundary conditions.
The GPS-gravimetry boundary value problem
Institute of Scientific and Technical Information of China (English)
YU; Jinhai; ZHANG; Chuanding
2005-01-01
How to determine the earth's external gravity field with the accuracy of O(T2) by making use of GPS data and gravity values measured on the earth's surface is dealt with in this paper. There are two main steps: to extend these measured values on the earth's surface onto the reference ellipsoid at first and then to seek for the integral solution of the external Neumann problem outside the ellipsoid. In addition, the corresponding judging criteria of accuracy to solve the GPS-gravity boundary value problem are established. The integral solution given in the paper not only contains all frequency-spectral information of the gravity field with the accuracy of O(T2),but is also easily computed. In fact, the solution has great significance for both theory and practice.
Complementary Lidstone Interpolation and Boundary Value Problems
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Ravi P. Agarwal
2009-01-01
Full Text Available We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial P2m(t of degree 2m, which involves interpolating data at the odd-order derivatives. For P2m(t we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a (2m+1th order differential equation and the complementary Lidstone boundary conditions.
Spectral integration of linear boundary value problems
Viswanath, Divakar
2012-01-01
Spectral integration is a method for solving linear boundary value problems which uses the Chebyshev series representation of functions to avoid the numerical discretization of derivatives. It is occasionally attributed to Zebib (J. of Computational Physics vol. 53 (1984), p. 443-455) and more often to Greengard (SIAM J. on Numerical Analysis vol. 28 (1991), p. 1071-1080). Its advantage is believed to be its relative immunity to errors that arise when nearby grid points are used to approximate derivatives. In this paper, we reformulate the method of spectral integration by changing it in four different ways. The changes consist of a more convenient integral formulation, a different way to treat and interpret boundary conditions, treatment of higher order problems in factored form, and the use of piecewise Chebyshev grid points. Our formulation of spectral integration is more flexible and powerful as show by its ability to solve a problem that would otherwise take 8192 grid points using only 96 grid points. So...
Spherical gravitational curvature boundary-value problem
Šprlák, Michal; Novák, Pavel
2016-08-01
Values of scalar, vector and second-order tensor parameters of the Earth's gravitational field have been collected by various sensors in geodesy and geophysics. Such observables have been widely exploited in different parametrization methods for the gravitational field modelling. Moreover, theoretical aspects of these quantities have extensively been studied and well understood. On the other hand, new sensors for observing gravitational curvatures, i.e., components of the third-order gravitational tensor, are currently under development. As the gravitational curvatures represent new types of observables, their exploitation for modelling of the Earth's gravitational field is a subject of this study. Firstly, the gravitational curvature tensor is decomposed into six parts which are expanded in terms of third-order tensor spherical harmonics. Secondly, gravitational curvature boundary-value problems defined for four combinations of the gravitational curvatures are formulated and solved in spectral and spatial domains. Thirdly, properties of the corresponding sub-integral kernels are investigated. The presented mathematical formulations reveal some important properties of the gravitational curvatures and extend the so-called Meissl scheme, i.e., an important theoretical framework that relates various parameters of the Earth's gravitational field.
Mixed Boundary Value Problem on Hypersurfaces
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R. DuDuchava
2014-01-01
Full Text Available The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation divC(A∇Cφ=f on a smooth hypersurface C with the boundary Γ=∂C in Rn. A(x is an n×n bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts Γ=ΓD∪ΓN and on ΓD the Dirichlet boundary conditions are prescribed, while on ΓN the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to divS(A∇S is proved, which is interpreted as the invertibility of this operator in the setting Hp,#s(S→Hp,#s-2(S, where Hp,#s(S is a subspace of the Bessel potential space and consists of functions with mean value zero.
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彭丽
2002-01-01
The finite element solution of two points boundary value problem for nonlinear ordinary differential equation is studied by using the collocation-Galerkin method.The Jacobi points are introduced to establish high orders of accuracy for the approximate solution.Numerical results are presented for a sample problem.
Boundary Value Problems for Boussinesq Type Systems
Energy Technology Data Exchange (ETDEWEB)
Fokas, A. S. [Cambridge University, Department of Applied Mathematics and Theoretical Physics (United Kingdom)], E-mail: t.fokas@damtp.cam.ac.uk; Pelloni, B. [University of Reading, Department of Mathematics (United Kingdom)], E-mail: b.pelloni@rdg.ac.uk
2005-02-15
We characterise the boundary conditions that yield a linearly well posed problem for the so-called KdV-KdV system and for the classical Boussinesq system. Each of them is a system of two evolution PDEs modelling two-way propagation of water waves. We study these problems with the spatial variable in either the half-line or in a finite interval. The results are obtained by extending a spectral transform approach, recently developed for the analysis of scalar evolution PDEs, to the case of systems of PDEs.The knowledge of the boundary conditions that should be imposed in order for the problem to be linearly well posed can be used to obtain an integral representation of the solution. This knowledge is also necessary in order to conduct numerical simulations for the fully nonlinear systems.
ON THE EXISTENCE AND UNIQUENESS OFSOLUTIONS FOR 2nTH-ORDER BOUNDARY VALUE PROBLEMS
Institute of Scientific and Technical Information of China (English)
PeiMinghe; SungKagChang
2005-01-01
In this paper, by using the Leray-Schauder continuation theorem, we establish the existence and uniqueness theorems of solutions of two-point boundary value problems for 2nth-order nonlinear differential equations with nonlinear growth.
Topological structures of boundary value problems in block elements
Babeshko, V. A.; Evdokimova, O. V.; Babeshko, O. M.
2016-10-01
Block structures are considered; a boundary value problem for a system of inhomogeneous partial differential equations with constant coefficients is formulated in each block of a structure. The problem of matching solutions to boundary value problems in blocks with each other by topological study of the properties of solutions in the block structure is examined in the conditions of correct solvability of boundary value problems in blocks of the block structure. Some new properties of solutions to boundary value problems in block structures are found that are important for applications.
2n阶非线性微分方程的第二边值问题%Second Boundary Value Problems for 2n-th Order Nonlinear Differential Equations
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吕显瑞; 徐庆; 高广学
2002-01-01
Sufficient conditions for the existence and uniqueness of second boundary value problems of two kinds of even order nonlinear differential equations are obtained. The proofs are based on the lemma on bilinear form, developed by A.C.Lazer, Schauder fixed point theorem and the Leray-Schauder degree theory, respectively.
BOUNDARY VALUE PROBLEMS, PARTIAL DIFFERENTIAL EQUATIONS ), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), (*NUMERICAL ANALYSIS, BOUNDARY VALUE PROBLEMS), FUNCTIONS(MATHEMATICS), DIFFERENCE EQUATIONS
A Kind of Boundary Value Problem for Hypermonogenic Function Vectors
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He Ju YANG; Yong Hong XIE; Yu Ying QIAO
2011-01-01
By the Plemelj formula and the compressed fixed point theorem, this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis. And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.
A boundary value problem for hypermonogenic functions in Clifford analysis
Institute of Scientific and Technical Information of China (English)
QIAO; Yuying
2005-01-01
This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.
Two new algorithms for discrete boundary value problems
Directory of Open Access Journals (Sweden)
Ravi P. Agarwal
1990-01-01
Full Text Available We propose two new methods of constructing the solutions of linear multi-point discrete boundary value problems. These methods are applied to solve some continuous two-point boundary value problems which are known to be numerically unstable.
Initial and boundary value problems for partial functional differential equations
Ntouyas , S. K.; P. Ch Tsamatos
1997-01-01
In this paper we study the existence of solutions to initial and boundary value problems of partial functional differential equations via a fixed-point analysis approach. Using the topological transversality theorem we derive conditions under which an initial or a boundary value problem has a solution.
THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMILINEAR ELLIPTIC EQUATION
Institute of Scientific and Technical Information of China (English)
Mo Jiaqi; Yao Jingsun
2001-01-01
The singularly perturbed boundary value problems for the semilinear elliptic equation are considered.Under suitable conditions and by using the fixed point theorem,the existence,uniqueness and asymptotic behavior of solution for the boundary value problems are studied.
Partial differential equations IX elliptic boundary value problems
Egorov, Yu; Shubin, M
1997-01-01
This EMS volume gives an overview of the modern theory of elliptic boundary value problems. The contribution by M.S. Agranovich is devoted to differential elliptic boundary problems, mainly in smooth bounded domains, and their spectral properties. This article continues his contribution to EMS 63. The contribution by A. Brenner and E. Shargorodsky concerns the theory of boundary value problems for elliptic pseudodifferential operators. Problems both with and without the transmission property, as well as parameter-dependent problems are considered. The article by B. Plamenevskij deals with general differential elliptic boundary value problems in domains with singularities.
Pseudo almost periodic solutions to parabolic boundary value inverse problems
Institute of Scientific and Technical Information of China (English)
2008-01-01
We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.
State-dependent impulses boundary value problems on compact interval
Rachůnková, Irena
2015-01-01
This book offers the reader a new approach to the solvability of boundary value problems with state-dependent impulses and provides recently obtained existence results for state dependent impulsive problems with general linear boundary conditions. It covers fixed-time impulsive boundary value problems both regular and singular and deals with higher order differential equations or with systems that are subject to general linear boundary conditions. We treat state-dependent impulsive boundary value problems, including a new approach giving effective conditions for the solvability of the Dirichlet problem with one state-dependent impulse condition and we show that the depicted approach can be extended to problems with a finite number of state-dependent impulses. We investigate the Sturm–Liouville boundary value problem for a more general right-hand side of a differential equation. Finally, we offer generalizations to higher order differential equations or differential systems subject to general linear boundary...
Initial value methods for boundary value problems
Meyer, Gunter H
1973-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
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徐云滨; 郑连存
2008-01-01
A class of singular nonlinear boundary value problems arising in the boundary layer behind expansion wave are studied. Sufficient conditions for the existence and uniqueness of positive solutions to the problems are established by utilizing the monotonic approaching technique. And a theoretical estimate formula for skin friction coefficient is presented. The numerical solution is presented by using the shoot method. The reliability and efficiency of the theoretical prediction are verified by numerical results.
EXISTENCE AND ITERATION OF POSITIVE SYMMETRIC SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we consider the existence of symmetric solutions to a nonlinear second order multi-point boundary value problem,and establish corresponding iterative schemes based on the monotone iterative method.
SOLVABILITY FOR FRACTIONAL-ORDER TWO-POINT BOUNDARY VALUE PROBLEM AT RESONANCE
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper, we are concerned with the existence of solution to a boundary value problem of nonlinear fractional differential equation at resonance. By means of the coincidence degree theory, the existence of solution is obtained.
ON THE EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR NEUMANN BOUNDARY VALUE PROBLEMS
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SunYan; XuBenlong; SunYongping
2005-01-01
By using fixed point index theory, we consider the existence of positive solutions for singular nonlinear Neumann boundary value problems. Our main results extend and improve many known results even for non-singular cases.
Existence results for anisotropic discrete boundary value problems
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Avci Avci
2016-06-01
Full Text Available In this article, we prove the existence of nontrivial weak solutions for a class of discrete boundary value problems. The main tools used here are the variational principle and critical point theory.
Positive Solutions for Higher Order Singular -Laplacian Boundary Value Problems
Indian Academy of Sciences (India)
Guoliang Shi; Junhong Zhang
2008-05-01
This paper investigates $2m-\\mathrm{th}(m≥ 2)$ order singular -Laplacian boundary value problems, and obtains the necessary and sufficient conditions for existence of positive solutions for sublinear 2-th order singular -Laplacian BVPs on closed interval.
BOUNDARY VALUE PROBLEM FOR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATION
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无
2006-01-01
The author employs the method of upper and lower solutions together with the monotone iterative technique to obtain the existence theorem of minimal and maximal solutions for a boundary value problem of second order impulsive differential equation.
Reconsideration on Homogeneous Quadratic Riemann Boundary Value Problem
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Lu Jian-ke
2004-01-01
The homogeneous quadratic Riemann boundary value problem (1) with Hǒlder continuous coefficients for the normal case was considered by the author in 1997. But the solutions obtained there are incomplete. Here its general method of solution is obtained.
Boundary value problems and Fourier expansions
MacCluer, Charles R
2004-01-01
Based on modern Sobolev methods, this text for advanced undergraduates and graduate students is highly physical in its orientation. It integrates numerical methods and symbolic manipulation into an elegant viewpoint that is consonant with implementation by digital computer. The first five sections form an informal introduction that develops students' physical and mathematical intuition. The following section introduces Hilbert space in its natural environment, and the next six sections pose and solve the standard problems. The final seven sections feature concise introductions to selected topi
Boundary value problems and Markov processes
Taira, Kazuaki
2009-01-01
This volume is devoted to a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called a Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel' boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by jumps and continuously in the state space and it obeys the Ventcel' boundary condition, which consists of six terms corresponding to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (or viscosity) phenomenon, the jump phenomenon on the boundary, and the inward jump phenomenon from the boundary. In particular, second-order elliptic differential operators are called diffusion operators and describe analytically strong Markov pr...
A nonlinear wave equation with a nonlinear integral equation involving the boundary value
Directory of Open Access Journals (Sweden)
Thanh Long Nguyen
2004-09-01
Full Text Available We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t}=0,quad xin Omega =(0,1,; 0
The Boundary Value Problem for Elliptic Equation in the Corner Domain
Zhidkov, E P
2000-01-01
This work is devoted to the studies of the solution behavior of the boundary value problem for a nonlinear elliptic equation in the corner domain. The formulation of the boundary value problem arises in magnitostatics when finding the magnetic field distribution by the method of two scalar potentials in the domain comprising ferromagnetic and vacuum. The problem nonlinearity is stipulated by the dependence of the medium properties (magnetic permeability) on the solution to be found. In connection with that the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirments, the corresponding solution of the boundary value problem will have a limited gradient.
Uniqueness and existence for bounded boundary value problems
Ehme, J.; Lanz, A.
2006-01-01
The existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established. We assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems. (c) 2005 Elsevier Inc. All righ
EXISTENCE OF SOLUTIONS TO A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,we study the existence of solutions to a third-order three-point boundary value problem.By imposing certain restrictions on the nonlinear term,we prove the existence of at least one solution to the boundary value problem by the method of lower and upper solutions.We are interested in the construction of lower and upper solutions.
Positive solutions of three-point boundary value problems
Institute of Scientific and Technical Information of China (English)
MIAO Ye-hong; ZHANG Ji-hui
2008-01-01
In this paper,we consider existence of single or multiple positive solutions of three-point boundary value problems involving one-dimensional p-Laplacian.We then study existence of solutions when the problems are in resonance cases.The proposed approach is based on the Krasnoselskii's fixed point theorem and the coincidence degree.
RIEMANN BOUNDARY VALUE PROBLEMS WITH GIVEN PRINCIPAL PART
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Li Weifeng; Du Jinyuan
2009-01-01
In this article, the authors discuss the Riemann boundary value problems with given principal part. First, authors consider a special case and give a classification of the solution class Rn by the way. And then, they consider the general case. The solvable conditions for this problem and its solutions is obtained when it is solvable.
Free Boundary Value Problems for Abstract Elliptic Equations and Applications
Institute of Scientific and Technical Information of China (English)
Veli SHAKHMUROV
2011-01-01
The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract Lp-spaces are given.In application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.
Solvability for fractional order boundary value problems at resonance
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Hu Zhigang
2011-01-01
Full Text Available Abstract In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation D 0 + α x ( t = f ( t , x ( t , x ′ ( t , x ″ ( t , t ∈ [ 0 , 1 ] , x ( 0 = x ( 1 , x ′ ( 0 = x ″ ( 0 = 0 , where D 0 + α denotes the Caputo fractional differential operator of order α, 2 < α ≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained. Mathematics Subject Classification (2000: 34A08, 34B15.
Directory of Open Access Journals (Sweden)
Zhiyong Wang
2008-09-01
Full Text Available In this paper, we study the existence of positive solutions for the nonlinear nth-order with m-point singular boundary-value problem. By using the fixed point index theory and a new fixed point theorem in cones, the existence of countably many positive solutions for a nonlinear singular boundary value problem are obtained.
Numerical methods for stiff systems of two-point boundary value problems
Flaherty, J. E.; Omalley, R. E., Jr.
1983-01-01
Numerical procedures are developed for constructing asymptotic solutions of certain nonlinear singularly perturbed vector two-point boundary value problems having boundary layers at one or both endpoints. The asymptotic approximations are generated numerically and can either be used as is or to furnish a general purpose two-point boundary value code with an initial approximation and the nonuniform computational mesh needed for such problems. The procedures are applied to a model problem that has multiple solutions and to problems describing the deformation of thin nonlinear elastic beam that is resting on an elastic foundation.
Timergaliev, S. N.; Kharasova, L. S.
2016-11-01
Solvability of one system of nonlinear second order partial differential equations with given initial conditions is considered in an arbitrary field. Reduction of the initial system of equations to one nonlinear operator equation is used to study the problem. The solvability is established with the use of the principle of contracting mappings. The method used in these studies is based on the integral representations for the displacements. These representations are constructed with the use of general solutions to the inhomogeneous Cauchy-Riemann equation.
EXISTENCE OF SOLUTIONS TO 2m-ORDER PERIODIC BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper, we are concerned with the existence of nontrivial solutions to a 2m-order nonlinear periodic boundary value problem. By the infinite dimensional Morse theory, under some conditions on nonlinear term, we obtain that there exist at least two nontrivial solutions.
Directory of Open Access Journals (Sweden)
Yigang Sun
2008-02-01
Full Text Available We establish the existence of multiple positive solutions for a singular nonlinear third-order periodic boundary value problem. We are mainly interested in the semipositone case. The proof relies on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique.
Non-Homogeneous Riemann Boundary Value Problem with Radicals
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The solution of the non-homogeneous Riemann boundary value problem with radicals (1.2)together with its condition of solvability is obtained for arbitrary positive integers p and q, which generalizes the results for the case p=q=2.
Solvability for fractional order boundary value problems at resonance
Hu Zhigang; Liu Wenbin
2011-01-01
Abstract In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation D 0 + α x ( t ) = f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) ) , t ∈ [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ′ ( 0 ) = x ″ ( 0 ) = 0 , where D 0 + α denotes the Caputo fractional differential o...
Fourth-order discrete anisotropic boundary-value problems
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Maciej Leszczynski
2015-09-01
Full Text Available In this article we consider the fourth-order discrete anisotropic boundary value problem with both advance and retardation. We apply the direct method of the calculus of variations and the mountain pass technique to prove the existence of at least one and at least two solutions. Non-existence of non-trivial solutions is also undertaken.
MULTIPLE SOLUTIONS TO AN ASYMPTOTICALLY LINEAR ROBIN BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for the case of the so-called resonance at infinity.
Fourth-order discrete anisotropic boundary-value problems
2015-01-01
In this article we consider the fourth-order discrete anisotropic boundary value problem with both advance and retardation. We apply the direct method of the calculus of variations and the mountain pass technique to prove the existence of at least one and at least two solutions. Non-existence of non-trivial solutions is also undertaken.
An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics
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A. Barari
2008-01-01
Full Text Available Variational iteration method (VIM is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM. The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.
Institute of Scientific and Technical Information of China (English)
李小龙
2012-01-01
The existence of positive solutions for nonlinear Robin boundary value problem in an ordered Banach spaces was discussed. An existence result of positive solutions was obtained by employing a new estimate of noncompactness measure and the fixed point index theory of condensing mapping.%讨论有序Banach空间E中的非线性Robin边值问题正解的存在性,通过非紧性测度的估计技巧与凝聚映射的不动点指数理论获得该问题正解的存在性结果.
Numerical Analysis of Forth-Order Boundary Value Problems in Fluid Mechanics and Mathematics
DEFF Research Database (Denmark)
Hosseinzadeh, E.; Barari, Amin; Fouladi, F.
2011-01-01
In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed...
Numerical analysis of fourth-order boundary value problems in fluid mechanics and mathematics
DEFF Research Database (Denmark)
Hosseinzadeh, Elham; Barari, Amin; Fouladi, Fama
2010-01-01
In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed...
Numerical Analysis of Forth-Order Boundary Value Problems in Fluid Mechanics and Mathematics
DEFF Research Database (Denmark)
Hosseinzadeh, E.; Barari, Amin; Fouladi, F.
2011-01-01
In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed...
Solvability of Boundary Value Problem at Resonance for Third-Order Functional Differential Equations
Indian Academy of Sciences (India)
Pinghua Yang; Zengji Du; Weigao Ge
2008-05-01
This paper is devoted to the study of boundary value problem of third-order functional differential equations. We obtain some existence results for the problem at resonance under the condition that the nonlinear terms is bounded or generally unbounded. In this paper we mainly use the topological degree theory.
EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS TO THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The existence and multiplicity of positive solutions to a periodic boundary value problem for nonlinear third-order ordinary differential equation are established, based on the zero point theorem concerning cone expansion and compression of order type. Our main approach is different from the previous papers on the existence of multiple positive solutions to the similar problem.
A boundary value problem for the wave equation
Directory of Open Access Journals (Sweden)
Nezam Iraniparast
1999-01-01
Full Text Available Traditionally, boundary value problems have been studied for elliptic differential equations. The mathematical systems described in these cases turn out to be “well posed”. However, it is also important, both mathematically and physically, to investigate the question of boundary value problems for hyperbolic partial differential equations. In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of special interest. The most recent works in this area have resulted in a number of interesting discoveries [3, 4, 5, 7, 8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.
Positive solutions and eigenvalues of nonlocal boundary-value problems
Directory of Open Access Journals (Sweden)
Jifeng Chu
2005-07-01
Full Text Available We study the ordinary differential equation $x''+lambda a(tf(x=0$ with the boundary conditions $x(0=0$ and $x'(1=int_{eta}^{1}x'(sdg(s$. We characterize values of $lambda$ for which boundary-value problem has a positive solution. Also we find appropriate intervals for $lambda$ so that there are two positive solutions.
Eigenvalues of boundary value problems for higher order differential equations
Wong, Patricia J. Y.; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Eigenvalues of boundary value problems for higher order differential equations
Patricia J. Y. Wong; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Noncontinuous data boundary value problems for Schr(o)dinger equation in Lipschitz domains
Institute of Scientific and Technical Information of China (English)
TAO Xiangxing
2006-01-01
The noncontinuous data boundary value problems for Schr(o)dinger equations in Lipschitz domains and its progress are pointed out in this paper.Particularly,the Lp boundary value problems with p＞1,and Hp boundary value problems with p＜1 have been studied.Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.
Numerical solution of fuzzy boundary value problems using Galerkin method
Indian Academy of Sciences (India)
SMITA TAPASWINI; S CHAKRAVERTY; JUAN J NIETO
2017-01-01
This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the proposed method, an application problem related to heat conduction has also been studied. The results obtained by the proposed methods are compared with the exact solution and other existing methods for demonstrating the validity and efficiency of the present method.
Free boundary value problems for a class of generalized diffusion equation
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied. Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function, convection function, heat flux function, and power law parameters by utilizing the shooting technique. The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.
Higher-order Lidstone boundary value problems for elliptic partial differential equations
Wang, Yuan-Ming
2005-08-01
The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.
Solvability of a class of second-order quasilinear boundary value problems
Institute of Scientific and Technical Information of China (English)
Qing-liu YAO
2009-01-01
The second-order quasilinear boundary value problems are considered when the nonlinear term is singular and the limit growth function at the infinite exists. With the introduction of the height function of the nonlinear term on a bounded set and the consideration of the integration of the height function, the existence of the solution is proven. The existence theorem shows that the problem has a solution ff the integration of the limit growth function has an appropriate value.
Institute of Scientific and Technical Information of China (English)
KANG Ping; YAO Jianli
2009-01-01
In this paper, we investigate the existence of symmetric solutions of singular nonlocal boundary value problems for systems of differential equations. Our analysis relies on a nonlinear alternative of Leray - schauder type. Our results presented here unify, generalize and significantly improve many known results in the literature.
Solvability of 2n-order m-point boundary value problem at resonance
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The existence of solutions for the 2n-order m-point boundary value problem at resonance is obtained by using the coincidence degree theory of Mawhin.We give an example to demonstrate our result.The interest is that the nonlinear term may be noncontinuous.
Directory of Open Access Journals (Sweden)
Mohamed Jleli
2014-01-01
Full Text Available A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper investigates the existence of positive solutions to systems of second order nonlocal boundary value problems with first order derivatives, in which the nonlinear term is not required to be continuous and involves first order derivatives. The main tool used in this paper is a fixed point index theory in a cone.
Positive solutions of multi-point boundary value problem of fractional differential equation
Directory of Open Access Journals (Sweden)
De-xiang Ma
2015-07-01
Full Text Available By means of two fixed-point theorems on a cone in Banach spaces, some existence and multiplicity results of positive solutions of a nonlinear fractional differential equation boundary value problem are obtained. The proofs are based upon some properties of Green’s function, which are also the key of the paper.
EXISTENCE OF POSITIVE SOLUTIONS TO SINGULAR SUBLINEAR SEMIPOSITONE NEUMANN BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The existence of positive solutions to a singular sublinear semipositone Neumann boundary value problem is considered. In this paper,the nonlinearity term is not necessary to be bounded from below and the function q(t) is allowed to be singular at t = 0 and t = 1.
Partial differential equations and boundary-value problems with applications
Pinsky, Mark A
2011-01-01
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems-rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate th
On Systems of Boundary Value Problems for Differential Inclusions
Institute of Scientific and Technical Information of China (English)
Lynn ERBE; Christopher C. TISDELL; Patricia J. Y. WONG
2007-01-01
Herein we consider the existence of solutions to second-order,two-point boundary value problems (BVPs) for systems of ordinary differential inclusions.Some new Bernstein –Nagumo condi-tions are presented that ensure a priori on the derivative of solutions to the differential inclusion.These a priori results are then applied,in conjunction with appropriate topological methods,to prove some new existence theorems for solutions to systems of BVPs for differential inclusions.The new conditions allow of the treatment of systems of BVPs without growth restrictions.
Fractional Extensions of some Boundary Value Problems in Oil Strata
Indian Academy of Sciences (India)
Mridula Garg; Alka Rao
2007-05-01
In the present paper, we solve three boundary value problems related to the temperature field in oil strata - the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros’ theorem is widely used.
Institute of Scientific and Technical Information of China (English)
陆海霞
2013-01-01
首先研究通过椭圆型偏微分方程歧点的连通分支的性质,然后得到椭圆型偏微分方程边值问题至少有一个正解存在结果.主要研究方法是全局分歧理论.%In this paper,the properties of the connected component containing the bifurcation points of a nonlinear elliptic partial differential equation is firstly studied.Then the existence of at least one positive solution for the boundary value problems of the equation is proved.The method to show our main results is the global bifurcation theol.
Chebyshev Finite Difference Method for Fractional Boundary Value Problems
Directory of Open Access Journals (Sweden)
Boundary
2015-09-01
Full Text Available This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivatives are described in the Caputo sense. Numerical results show that this method is of high accuracy and is more convenient and efficient for solving boundary value problems involving fractional ordinary differential equations. AMS Subject Classification: 34A08 Keywords and Phrases: Chebyshev polynomials, Gauss-Lobatto points, fractional differential equation, finite difference 1. Introduction The idea of a derivative which interpolates between the familiar integer order derivatives was introduced many years ago and has gained increasing importance only in recent years due to the development of mathematical models of a certain situations in engineering, materials science, control theory, polymer modelling etc. For example see [20, 22, 25, 26]. Most fractional order differential equations describing real life situations, in general do not have exact analytical solutions. Several numerical and approximate analytical methods for ordinary differential equation Received: December 2014; Accepted: March 2015 57 Journal of Mathematical Extension Vol. 9, No. 3, (2015, 57-71 ISSN: 1735-8299 URL: http://www.ijmex.com Chebyshev Finite Difference Method for Fractional Boundary Value Problems H. Azizi Taft Branch, Islamic Azad University Abstract. This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivative
Macías-Díaz, J E; 10.1002/zamm.200700172
2011-01-01
In this work, we present a conditionally stable finite-difference scheme that consistently approximates the solution of a general class of (3+1)-dimensional nonlinear equations that generalizes in various ways the quantitative model governing discrete arrays consisting of coupled harmonic oscillators. Associated with this method, there exists a discrete scheme of energy that consistently approximates its continuous counterpart. The method has the properties that the associated rate of change of the discrete energy consistently approximates its continuous counterpart, and it approximates both a fully continuous medium and a spatially discretized system. Conditional stability of the numerical technique is established, and applications are provided to the existence of the process of nonlinear supratransmission in generalized Klein-Gordon systems and the propagation of binary signals in semi-unbounded, three-dimensional arrays of harmonic oscillators coupled through springs and perturbed harmonically at the bound...
Green's function solution to spherical gradiometric boundary-value problems
Martinec, Z.
2003-05-01
Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {orr}, {or/,or5} and {o//mo55,o/5}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {or/,or5} and {o//mo55,o/5}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point `=0 is investigated for the component orr. The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1-11] show that the presented solution refines the former solution.
Partial differential equations & boundary value problems with Maple
Articolo, George A
2009-01-01
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple''s animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website. Ancillary list: Maple files- http://www.elsevierdirect.com/companion.jsp?ISBN=9780123747327 Provides a quick overview of the software w/simple commands needed to get startedIncludes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equationsIncorporates an early introduction to Sturm-L...
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
Alikhanov, A A
2011-01-01
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
A CLASS OF SINGULARLY PERTURBED ROBIN BOUNDARY VALUE PROBLEMS FOR SEMILINEAR ELLIPTIC EQUATION
Institute of Scientific and Technical Information of China (English)
MoJiaqi
2001-01-01
The singularly perturbed Robin boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem the existence ,uniqueness and asymptotic behavior of solution for the boundary value problems are studied.
An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems
Directory of Open Access Journals (Sweden)
Mohammad Maleki
2012-01-01
Full Text Available An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE. By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.
Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems
Directory of Open Access Journals (Sweden)
Bojan Orel
2014-01-01
Full Text Available A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval −1,1, we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010 and further analyzed by Orel and Perne (2012. The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation number N for the new class of Chebyshev-Fourier-collocation (CFC methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC methods.
The boundary value problem for discrete analytic functions
Skopenkov, Mikhail
2013-06-01
This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.
POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,we establish sufficient conditions for the existence of positive solutions to a general class of integral boundary value problem(BVP) of nonlinear fractional functional differential equation.A differential operator is taken in the RiemannLiouville sense.Our analysis relies on the Krasnosel'skii fixed-point theorem in cones.We also give examples to illustrate the applicability of our results.
Dirichlet-Neumann bracketing for boundary-value problems on graphs
Directory of Open Access Journals (Sweden)
Sonja Currie
2005-08-01
Full Text Available We consider the spectral structure of second order boundary-value problems on graphs. A variational formulation for boundary-value problems on graphs is given. As a consequence we can formulate an analogue of Dirichlet-Neumann bracketing for boundary-value problems on graphs. This in turn gives rise to eigenvalue and eigenfunction asymptotic approximations.
Institute of Scientific and Technical Information of China (English)
Li Ta-tsien(李大潜); Peng Yue-Jun
2003-01-01
Abstract We prove that the C0 boundedness of solution impliesthe global existence and uniqueness of C1 solution to the initial-boundary value problem for linearly degenerate quasilinear hyperbolic systems of diagonal form with nonlinear boundary conditions. Thus, if the C1 solution to the initial-boundary value problem blows up in a finite time, then the solution itself must tend to the infinity at the starting point of singularity.
Directory of Open Access Journals (Sweden)
A. H. Bhrawy
2014-01-01
Full Text Available We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials Jn(α,β(r with α,β∈(-1,∞, r∈(0,1 and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.
Finite element method for solving geodetic boundary value problems
Fašková, Zuzana; Čunderlík, Róbert; Mikula, Karol
2010-02-01
The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches, e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally, we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements as input data. All tests show qualitative and quantitative agreement with the given solutions.
Directory of Open Access Journals (Sweden)
Ishfaq Ahmad Ganaie
2014-01-01
Full Text Available Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.
Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues
Canada, Antonio
2009-01-01
This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.
Institute of Scientific and Technical Information of China (English)
侯祥林; 刘铁林; 翟中海
2011-01-01
针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法.%For elliptic nonlinear partial differential equations with boundary value problem, based on difference method and dynamic design variable optimization method, by taking unknown function value on discrete net point as design variables, difference equation of all the discrete net points is constructed as an objective function. A kind of optimization algorithm about solving unknown function value on discrete net point is proposed. Universal computing program is designed. Practical example is analyzed. By comparing the computing result with the analytical solution, effectiveness and feasibility are verified. Thus complicated nonlinear mathematical physics equations can be solved by the numerical calculation method.
Institute of Scientific and Technical Information of China (English)
Yi-rang YUAN; Chang-feng LI; Cheng-shun YANG; Yu-ji HAN
2009-01-01
The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the two-dimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal order l2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.
Pindza, Edson; Maré, Eben
2017-03-01
A modified discrete singular convolution method is proposed. The method is based on the single (SE) and double (DE) exponential transformation to speed up the convergence of the existing methods. Numerical computations are performed on a wide variety of singular boundary value and singular perturbed problems in one and two dimensions. The obtained results from discrete singular convolution methods based on single and double exponential transformations are compared with each other, and with the existing methods too. Numerical results confirm that these methods are considerably efficient and accurate in solving singular and regular problems. Moreover, the method can be applied to a wide class of nonlinear partial differential equations.
Directory of Open Access Journals (Sweden)
Xiangchao Meng
2017-07-01
Full Text Available The boundary value problem of deflections of vertical with ellipsoid boundary is studied in the paper. Based on spherical harmonic series, the ellipsoidal corrections for the boundary value problem are derived so that it can be well solved. In addition, an imitation arithmetic is given for examining the accuracies of solutions for the boundary value problem as well as its spherical approximation problem, and the computational results illustrate that the boundary value problem has higher accuracy than its spherical approximation problem if deflection of the vertical are measured on geoid.
Introduction to Mathematical Physics. Calculus of Variations and Boundary-value Problems
Adamyan, V. M.; Sushko, M. Ya.
2013-01-01
This book considers posing and the methods of solving simple linear boundary-value problems in classical mathematical physics. The questions encompassed include: the fundamentals of calculus of variations; one-dimensional boundary-value problems in the oscillation and heat conduction theories, with a detailed analysis of the Sturm-Liouville boundary-value problem and substantiation of the Fourier method; sample solutions of the corresponding problems in two and three dimensions, with essentia...
Institute of Scientific and Technical Information of China (English)
杜增吉; 林晓洁; 葛渭高
2005-01-01
This paper is concerned with the following second-order vector boundary value problem:x″=f(t,Sx,x,x′),0＜t＜1,x(0)=A,g(x(1),x′(1))=B,where x,f,g,A and B are n-vectors.Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.
Existence of Positive Solutions for Higher Order Boundary Value Problem on Time Scales
Institute of Scientific and Technical Information of China (English)
XIE DA-PENG; LIU YANG; SUN MING-ZHE; Li Yong
2013-01-01
In this paper,we investigate the existence of positive solutions of a class higher order boundary value problems on time scales.The class of boundary value problems educes a four-point (or three-point or two-point) boundary value problems,for which some similar results are established.Our approach relies on the Krasnosel'skii fixed point theorem.The result of this paper is new and extends previously known results.
The RH boundary value problem of the k-monogenic functions
Bu, Yude; Du, Jinyuan
2008-11-01
In this paper we study the Riemann and Hilbert problems of k-monogenic functions. By using Euler operator, we transform the boundary value problem of k-monogenic functions into the boundary value problems of monogenic functions. Then by the Almansi-type theorem of k-monogenic functions, we get the solutions of these problems.
Energy Technology Data Exchange (ETDEWEB)
Gazzaniga, G.; Sacchi, G. [Istituto di Analisi Numerica, Pavia (Italy)
1995-12-01
Different Domain Decomposition techniques for the solution of elliptic boundary-value problems are considered. The results of the implementation on a parallel distributed memory architecture are discussed.
Institute of Scientific and Technical Information of China (English)
袁益让
1994-01-01
The software for oil-gas transport and accumulation is to describe the history of oil-gas transport and accumulation in basin evolution. It is of great value in rational evaluation of prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary value problem. This paper puts forward a kind of characteristic finite difference schemes, and derives from them optimal order estimates in l~2 norm for the error in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, for model numerical method and for software development.
Vaneeva, O. O.; Papanicolaou, N. C.; Christou, M. A.; Sophocleous, C.
2014-09-01
The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.
Robin Boundary Value Problem for One-Dimensional Landau-Lifshitz Equations
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Shi Jin DING; Jin Rui HUANG; Xiao E LIU
2012-01-01
In this paper,we are concerned with the existence and uniqueness of global smooth solution for the Robin boundary value problem of Landau-Lifshitz equations in one dimension when the boundary value depends on time t.Furthermore,by viscosity vanishing approach,we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.
Higher order non-local (n-1,1) conjugate type boundary value problems
Webb, J. R. L.
2009-05-01
We show how some recent work of Webb and Infante, which gave a unified method of tackling many nonlocal boundary value problems, can be applied to some higher order boundary value problems with more general nonlocal boundary conditions than previously studied. This improves some recent work on problems with conjugate type boundary conditions.
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Chen Yi
2011-01-01
Full Text Available We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii's fixed point theorem are applied to establish the existence results.
Roul, Pradip; Warbhe, Ujwal
2017-08-01
The classical homotopy perturbation method proposed by J. H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999) is useful for obtaining the approximate solutions for a wide class of nonlinear problems in terms of series with easily calculable components. However, in some cases, it has been found that this method results in slowly convergent series. To overcome the shortcoming, we present a new reliable algorithm called the domain decomposition homotopy perturbation method (DDHPM) to solve a class of singular two-point boundary value problems with Neumann and Robin-type boundary conditions arising in various physical models. Five numerical examples are presented to demonstrate the accuracy and applicability of our method, including thermal explosion, oxygen-diffusion in a spherical cell and heat conduction through a solid with heat generation. A comparison is made between the proposed technique and other existing seminumerical or numerical techniques. Numerical results reveal that only two or three iterations lead to high accuracy of the solution and this newly improved technique introduces a powerful improvement for solving nonlinear singular boundary value problems (SBVPs).
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Gai Gongqi
2011-01-01
Full Text Available Abstract This article studies the boundary value problems for the third-order nonlinear singular difference equations Δ 3 u ( i - 2 + λ a ( i f ( i , u ( i = 0 , i ∈ [ 2 , T + 2 ] , satisfying five kinds of different boundary value conditions. This article shows the existence of positive solutions for positone and semi-positone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone. MSC [2008]: 34B15; 39A10.
Boundary-value problems for elliptic functional-differential equations and their applications
Skubachevskii, A. L.
2016-10-01
Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics. Bibliography: 137 titles.
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高承华
2014-01-01
In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [G. Bonanno and P. Candito, Appl. Anal., 88(4) (2009), pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and µ in some suitable intervals. The approaches we use are the critical point theory.
Initial and Boundary Value Problems for Two-Dimensional Non-hydrostatic Boussinesq Equations
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沈春; 孙梅娜
2005-01-01
Based on the theory of stratification, the weU-posedness of the initial and boundary value problems for the system of twodimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.
m-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATION AT RESONANCE
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无
2012-01-01
In his paper,we obtain a general theorem concerning the existence of solutions to an m-point boundary value problem for the second-order differential equation with impulses.Moreover,the result can also be applied to study the usual m-point boundary value problem at resonance without impulses.
EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A THREE-POINT BOUNDARY VALUE PROBLEM
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无
2012-01-01
In this paper, we are concerned with the existence and nonexistence of positive solutions to a three-point boundary value problems. By Krasnoselskii's fixed point theorem in Banach space, we obtain sufficient conditions for the existence and non-existence of positive solutions to the above three-point boundary value problems.
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Bao-qiang Yan; Donal O'Regan; Ravi P.Agarwal
2008-01-01
This paper discusses both the nonexistence of positive solutions for second-order three-point bound ary value problems when the nonlinear term f(t,x,y) is superlinear in y at y=0 and the existence of multiple positive solutions for second-order three-point boundary value problems when the nonlinear term f(t,x,y) is superlinear in x at +∞.
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无
2010-01-01
By constructing an explicit Green function and using the fixed point index theory on a cone, we present some existence results of positive solutions to a class of second-order singular semipositive Neumann boundary value problem, where the nonlinear term is allowed to be nonnegative and unbounded.
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无
2011-01-01
This paper deals with the existence of denumberable positive solutions to boundary value problems of delay differential equations with denumberable singularities on infinite intervals. By the fixed-point index theory and a new fixed-point theorem in cones, the existence of denumberable positive solutions is obtained under some suitable growth conditions imposed on the nonlinear term.
Positive solutions of some nonlocal boundary value problems
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Gennaro Infante
2003-01-01
employed. In particular, we do not require all the parameters occurring in the boundary conditions to be positive. Our results allow more general behaviour for the nonlinear term than being either sub- or superlinear.
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
In this paper,authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order.They first give the well posedness of general discontinuous boundary value problems,reduce the discontinuousboundary value problems to a variation problem,and then find the numerical solutions ofabove problem by the finite element method.Finally authors give some error-estimates of the foregoing numerical solutions.
Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth-Order Boundary Value Problems
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A. H. Bhrawy
2011-01-01
Full Text Available We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs. An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.
Existence of solutions to fractional boundary-value problems with a parameter
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Ya-Ning Li
2013-06-01
Full Text Available This article concerns the existence of solutions to the fractional boundary-value problem $$displaylines{ -frac{d}{dt} ig(frac{1}{2} {}_0D_t^{-eta}+ frac{1}{2}{}_tD_{T}^{-eta}igu'(t=lambda u(t+abla F(t,u(t,quad hbox{a.e. } tin[0,T], cr u(0=0,quad u(T=0. }$$ First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u, we show the existence of weak solutions of the problem when $lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.
Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials
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Muhammad Aslam Noor
2008-01-01
Full Text Available We apply the variational iteration method using He's polynomials (VIMHP for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
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崔国忠; 张志平; 等
2002-01-01
This paper deals with the initial boundary value value problem for the Boltzmann-Poisson system ,which arises in semiconductor physics,with absorbing boundary.The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.
THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMILINEAR ELLIPTIC EQUATION OF HIGHER ORDER
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Chen Songlin; Mo Jiaqi
2000-01-01
The singularly perturbed boundary value problems for the semilinear elliptic equation of higher order are considered. Under suitable conditions and by using the fixed point theoren the existence, uniqueness and asymp totic behavior of solution for the boundary value tproblems are studied.
Boundary value problems on the half line in the theory of colloids
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Ravi P. Agarwal
2002-01-01
Full Text Available We present existence results for some boundary value problems defined on infinite intervals. In particular our discussion includes a problem which arises in the theory of colloids.
Electromagnetic wave theory for boundary-value problems an advanced course on analytical methods
Eom, Hyo J
2004-01-01
Electromagnetic wave theory is based on Maxwell's equations, and electromagnetic boundary-value problems must be solved to understand electromagnetic scattering, propagation, and radiation. Electromagnetic theory finds practical applications in wireless telecommunications and microwave engineering. This book is written as a text for a two-semester graduate course on electromagnetic wave theory. As such, Electromagnetic Wave Theory for Boundary-Value Problems is intended to help students enhance analytic skills by solving pertinent boundary-value problems. In particular, the techniques of Fourier transform, mode matching, and residue calculus are utilized to solve some canonical scattering and radiation problems.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
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Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
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无
2001-01-01
This paper deals with boundary value problems for linear uniformly elliptic systems. First the general linear uniformly elliptic system of the first order equations is reduced to complex form, and then the compound boundary value problem for the complex equations of the first order is discussed. The approximate solutions of the boundary value problem are found by the variation-difference method, and the error estimates for the approximate solutions are derived.Finally the approximate method of the oblique derivative problem for linear uniformly elliptic equations of the second or der is introduced.
A Boundary Value Problem for Hermitian Monogenic Functions
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Reyes JuanBory
2008-01-01
Full Text Available Abstract We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in . Necessary and sufficient conditions for the solvability of this problem are obtained.
A coupled BEM-FEM method for finite strain magneto-elastic boundary-value problems
Nedjar, B.
2016-12-01
The first objective of this contribution is the formulation of nonlinear problems in magneto-elasticity involving finite geometry of the surrounding free space. More specifically for the magnetic part of the problem, the surrounding free space is described by means of a boundary integral equation for which boundary elements are used that are appropriately coupled with the finite element discretization used inside the material. The second objective is to develop a numerical strategy to solve the strongly coupled magneto-mechanics problem at hand. Herein we provide a staggered scheme consisting of a magnetostatic resolution employing the above coupled BEM-FEM procedure at fixed deformation, followed by a mechanical resolution at fixed magnetic fields. This decoupled method renders the whole solution strategy very appealing since, among others, the first BEM-FEM resolution is linear for some prototype models, and the remaining mechanical resolution is analogous to nowadays classical nonlinear elastostatic problems in the finite strain range. Some nonlinear boundary-value problems are simulated to demonstrate the applicability of the proposed framework.
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Chen Guowang; Xue Hongxia
2008-01-01
In this article, the existence, uniqueness and regularities of the global gener-alized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation utt -uxx-auxxtt+bux4 - duxxt= f(u)xx are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
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B. Juarez Campos
2016-01-01
Full Text Available We study the mixed initial-boundary value problem for the capillary wave equation: iut+u2u=∂x3/2u, t>0, x>0; u(x,0=u0(x, x>0; u(0,t+βux(0,t=h(t, t>0, where ∂x3/2u=(1/2π∫0∞signx-y/x-yuyy(y dy. We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
Periodic Boundary Value Problems for Functional Differential Equations%带滞后项的非线性微分方程的周期边值问题
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祁爱琴; 高丽; 张全信
2009-01-01
The periodic boundary value problems for nonlinear functional differential equa-tions was discussed.The existence of maximal and minimal solutions was obtained when the lower and upper solutions satisfied the formal or reverse order.
A Boundary Value Problem for Hermitian Monogenic Functions
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Ricardo Abreu Blaya
2008-02-01
Full Text Available We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in Ã¢Â„Âm,Ã¢Â€Â‰m=2n. Necessary and sufficient conditions for the solvability of this problem are obtained.
Laplace Boundary-Value Problem in Paraboloidal Coordinates
Duggen, L.; Willatzen, M.; Voon, L. C. Lew Yan
2012-01-01
This paper illustrates both a problem in mathematical physics, whereby the method of separation of variables, while applicable, leads to three ordinary differential equations that remain fully coupled via two separation constants and a five-term recurrence relation for series solutions, and an exactly solvable problem in electrostatics, as a…
Unbounded Periodic Solutions to Serrin's Overdetermined Boundary Value Problem
Fall, Mouhamed Moustapha; Minlend, Ignace Aristide; Weth, Tobias
2017-02-01
We study the existence of nontrivial unbounded domains {Ω} in RN such that the overdetermined problem {-Δ u = 1 quad in Ω}, quad u = 0, quad partial_{ν} u = const quad on partial Ω admits a solution u. By this, we complement Serrin's classification result from 1971, which yields that every bounded domain admitting a solution of the above problem is a ball in RN. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.
Contrast structure for singular singularly perturbed boundary value problem
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王爱峰; 倪明康
2014-01-01
The step-type contrast structure for a singular singularly perturbed problem is shown. By use of the method of boundary function, the formal asymptotic expansion is constructed. At the same time, based on sewing orbit smooth, the existence of the step-type solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of the present results.
Existence and uniqueness of solutions for a Neumann boundary-value problem
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Safia Benmansour
2011-09-01
Full Text Available In this article, we show the existence and uniqueness of positive solutions for perturbed Neumann boundary-value problems of second-order differential equations. We use a fixed point theorem for general $alpha$-concave operators.
EXISTENCE FOR SECOND-ORDER FOUR-POINT BOUNDARY VALUE PROBLEM AT RESONANCE
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无
2006-01-01
This paper is concerned with the existence of solutions for a second-order four-point boundary value problem at resonance. The main methods depend on the technique of the upper and lower solutions and the coincidence degree theory.
EXISTENCE OF TRIPLE POSITIVE SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM
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无
2011-01-01
We apply a fixed point theorem to verify the existence of at least three positive solutions to a multi-point boundary value problem with p-Laplacian. Existence criteria which ensure the existence of triple positive solutions are established.
LIMIT BEHAVIOUR OF SOLUTIONS TO EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS
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LI Fengquan
2002-01-01
In this paper, we discuss the limit behaviour of solutions to equivalued surface boundary value problem for parabolic equations when the equivalued surface boundary shrinks to a point and the space dimension of the domain is two or more.
INITIAL-BOUNDARY VALUE PROBLEM FOR THE LANDAU-LIFSHITZ SYSTEM WITH APPLIED FIELD
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Guo Boling; Ding Shijin
2000-01-01
In this paper, the existence and partial regularity of weak solution to the initial-boundary value problem of Landau-Lifshitz equations with applied fields in a 2D bounded domain are obtained by the penalty method.
Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems
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Moulay Rchid Sidi Ammi
2013-01-01
Full Text Available We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales.
MULTIPLE POSITIVE SOLUTIONS TO FOURTH-ORDER SINGULAR BOUNDARY VALUE PROBLEMS
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无
2011-01-01
In this paper,using the Krasnaselskii's fixed point theory in cones and localization method,under more general conditions,the existence of n positive solutions to a class of fourth-order singular boundary value problems is considered.
Singularities of mixed boundary value problems in electrical impedance tomography.
Pidcock, M; Ciulli, S; Ispas, S
1995-08-01
The importance of accurate mathematical modelling in the development of image reconstruction algorithms for electrical impedance tomography (EIT) has been discussed in a number of recent papers. It is particularly important in iterative reconstruction schemes where the forward problem of calculating the electric potential from Neumann boundary data is solved many times. One area which needs to be considered it the mathematical modelling of the electrodes used in the technique. In this paper we discuss one of the more sophisticated models which has been proposed and present the results of a number of numerical and analytic calculations which we have made as a contribution to the understanding of this question.
Asymptotic Solution of the Theory of Shells Boundary Value Problem
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I. V. Andrianov
2007-01-01
Full Text Available This paper provides a state-of-the-art review of asymptotic methods in the theory of plates and shells. Asymptotic methods of solving problems related to theory of plates and shells have been developed by many authors. The main features of our paper are: (i it is devoted to the fundamental principles of asymptotic approaches, and (ii it deals with both traditional approaches, and less widely used, new approaches. The authors have paid special attention to examples and discussion of results rather than to burying the ideas in formalism, notation, and technical details.
A symmetric solution of a multipoint boundary value problem at resonance
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We apply a coincidence degree theorem of Mawhin to show the existence of at least one symmetric solution of the nonlinear second-order multipoint boundary value problem u ″ ( t = f ( t , u ( t , | u ′ ( t | , t ∈ ( 0 , 1 , u ( 0 = ∑ i = 1 n μ i u ( ξ i , u ( 1 − t = u ( t , t ∈ ( 0 , 1 ] , where 0 < ξ 1 < ξ 2 < … ≤ ξ n 1 / 2 , ∑ i = 1 n μ i = 1 , f : [ 0 , 1 ] × ℝ 2 → ℝ with f ( t , x , y = f ( 1 − t , x , y , ( t , x , y ∈ [ 0 , 1 ] × ℝ 2 , satisfying the Carathéodory conditions.
Asymptotic behavior of elliptic boundary-value problems with some small coefficients
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Senoussi Guesmia
2008-04-01
Full Text Available The aim of this paper is to analyze the asymptotic behavior of the solutions to elliptic boundary-value problems where some coefficients become negligible on a cylindrical part of the domain. We show that the dimension of the space can be reduced and find estimates of the rate of convergence. Some applications to elliptic boundary-value problems on domains becoming unbounded are also considered.
INHOMOGENEOUS INITIAL-BOUNDARY VALUE PROBLEM FOR GINZBURG-LANDAU EQUATIONS
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杨灵娥; 郭柏灵; 徐海祥
2004-01-01
Some integral identities of smooth solution of inhomogeneous initial boundary value problem of Ginzburg-Landau equations were deduced, by which a priori estimates of the square norm on boundary of normal derivative and the square norm of partial derivatives were obtained. Then the existence of global weak solution of inhomogeneous initial-boundary value problem of Ginzburg-Landau equations was proved by the method of approximation technique and a priori estimates and making limit.
GLOBAL BEHAVIOR OF POSITIVE SOLUTIONS TO A KIND OF THREE-POINT BOUNDARY VALUE PROBLEMS
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无
2009-01-01
In this paper, using the fixed-point index theorems and the cone theory we study the structure of the positive solutions to a kind of second-order three-point boundary value problems. We obtain the result which guarantee that the positive solution set of the three-point boundary value problem has a continuum, which means that there exists a nonempty, closed and connected subset.
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Ruihui Huang
2013-01-01
Full Text Available We study the existence and uniqueness of solutions for a class of antiperiodic boundary value problems of the fractional differential equation with a p-Laplacian operator. Based on the Leray-Schauder nonlinear alternative, several sufficient conditions of the existence and uniqueness of solution of the above problem are established. Our results improve and complement the recent work of Chen and Liu, 2012.
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Javed Ali
2012-01-01
Full Text Available We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM. The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM, variational iteration (VIM, homotopy perturbation (HPM, and variational iteration decomposition method (VIDM. The results show that the proposed method is more effective and reliable.
Numerical investigation of the boundary value problem for the carleman system of equations
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Vasilyeva Ol’ga Aleksandrovna
2016-12-01
Full Text Available The boundary value problem for the Carleman system of equations is considered. The problem is investigated numerically for initial conditions which are perturbed nonnegative stationary solutions of the problem. First point of the paper is numerical investigation of solution of the boundary value problem with perturbed positive stationary solutions as an initial condition. The time dependence of the maximum deviation of the solution of the stationary solution problem of stationary solutions is investigated. The results of numerical problem solution are presented. The time dependence of the energy of perturbations of stationary solutions of the problem is presented. The solution stabilization to the stationary solution problem is obtained. The solution stabilization time is compared with stabilization time in periodic case. Second point of the paper is numerical investigation of solution of the boundary-value problem with perturbed zero stationary solutions as an initial condition. The results of numerical problem solution are presented.
Wavelets, turbulence, and boundary value problems for partial differential equations
Weiss, John E.
1995-04-01
In this paper the qualitative properties of an inviscid, incompressible two-dimensional fluid are examined by numerical methods based on the compactly supported wavelets (the wavelet- Galerkin method). In particular, we examine the behavior of the spatial gradients of the vorticity. The growth of these gradients is related to the transfer of enstrophy (integral of squared vorticity) to the small-scales of the fluid motion. Implicit time differencing and wavelet-Galerkin space discretization allow a direct investigation of the long time behavior of the inviscid fluid. The effects of hyperviscosity on the long time limit are examined. To solve boundary problems we developed a new numerical method for the solution of partial differential equations in nonseparable domains. The method uses a wavelet-Galerkin solver with a nontrivial adaptation of the standard capacitance matrix method. The numerical solutions exhibit spectral convergence with regard to the order of the compactly supported, Daubechies wavelet basis. Furthermore, the rate of convergence is found to be independent of the geometry. We solve the Helmholtz equation since, for the indefinite case, the solutions have qualitative properties that well illustrate the applications of our method.
A Kind of Boundary Element Methods for Boundary Value Problem of Helmholtz Equation
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张然; 姜正义; 马富明
2004-01-01
Problems for electromagnetic scattering are of significant importance in many areas of technology. In this paper we discuss the scattering problem of electromagnetic wave incident by using boundary element method associated with splines. The problem is modelled by a boundary value problem for the Helmholtz eouation
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Andrei V. Obukhovskiĭ
2003-05-01
Full Text Available We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained.
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Jing Bao YANG; Zhong Li WEI
2011-01-01
By employing the fixed point theorem of cone expansion and compression of norm type,we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter.Some sufficient conditions for the existence of positive solutions are established.In the last section,an example is presented to illustrate the applications of our main results.
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Zhang Peiguo
2011-01-01
Full Text Available Abstract By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator. MSC: 34B15; 34B16.
Zhu, C
2003-01-01
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation. To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in Evans (1998 Partial Differential Equations (Providence, RI: American Mathematical Society) and Hopf (1950 Commun. Pure Appl. Math. 3 201-30), give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation.
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Zhu, Changjiang; Duan, Renjun [Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 430079, People' s Republic of China (China)
2003-02-28
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation. To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in Evans (1998 Partial Differential Equations (Providence, RI: American Mathematical Society) and Hopf (1950 Commun. Pure Appl. Math. 3 201-30), give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation.
Discrete approximations for singularly perturbed boundary value problems with parabolic layers
Farrell, P.A.; Hemker, P.W.; Shishkin, G.I.
1995-01-01
Singularly perturbed boundary value problems for equations of elliptic and parabolic type are studied. For small values of the perturbation parameter, parabolic boundary layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and th
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
In this paper,we study the existence and approximation of solution to boundary value problems for impulsive differential equations with delayed arguments.Sufficient conditions are established for the existence of a unique solution or extremal ones to the given problem.A monotone iterative technique is applied.
On the Solutions of Some Boundary Value Problems for the General Kdv Equation
Energy Technology Data Exchange (ETDEWEB)
Ignatyev, M. Yu., E-mail: mikkieram@gmail.com [Saratov State University, Department of Mathematics (Russian Federation)
2014-12-15
This paper is concerned with a class of partial differential equations, which are linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary conditions of a certain special form is studied. We construct some class of solutions of the problem using the inverse spectral method.
EXISTENCE THEOREM ABOUT MULTIPLE POSITIVE SOLUTIONS TO p-LAPLACIAN BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,we apply a fixed point theorem to verify the existence of multiple positive solutions to a p-Laplacian boundary value problem.Sufficient conditions are established which guarantee the existence of multiple positive solutions to the problem.
Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem
Olgar, Hayati; Mukhtarov, Oktay; Aydemir, Kadriye
2016-08-01
We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piece-wise continuous potential together with eigenparameter-dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular it is shown that the generalized eigen-functions form a Riesz basis of the adequate Hilbert space.
ON SOLUTION OF A KIND OF RIEMANN BOUNDARY VALUE PROBLEM WITH SQUARE ROOTS
Institute of Scientific and Technical Information of China (English)
路见可
2002-01-01
Solution of the Riemann boundary value problem with square roots (1.1)for analytic functions proposed in [1] is reconsidered, which was solved under certain assumptions on the branch points appeared. Here, the work is continued without these assumptions and the problem is solved in the general case.
Well-posedness of boundary-value problems for partial differential equations of even order
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Djumaklych Amanov
2014-04-01
Full Text Available In this article, we establish the well-posedness of two boundary value problems for 2k-th order partial differential equations. It is shown that the solvability of these problems depends on the evenness and oddness of the number k.
Well-posedness of boundary-value problems for partial differential equations of even order
Djumaklych Amanov; Allaberen Ashyralyev
2014-01-01
In this article, we establish the well-posedness of two boundary value problems for 2k-th order partial differential equations. It is shown that the solvability of these problems depends on the evenness and oddness of the number k.
Adaptation of a two-point boundary value problem solver to a vector-multiprocessor environment
Energy Technology Data Exchange (ETDEWEB)
Wright, S.J. (Mathematics Dept., North Carolina State Univ., Raleigh, NC (US)); Pereyra, V. (Weidlinger Associates, Los Angeles, CA (US))
1990-05-01
Systems of linear equations arising from finite-difference discretization of two-point boundary value problems have coefficient matrices that are sparse, with most or all of the nonzeros clustered in blocks near the main diagonal. Some efficiently vectorizable algorithms for factorizing these types of matrices and solving the corresponding linear systems are described. The relative effectiveness of the different algorithms varies according to the distribution of initial, final, and coupled end conditions. The techniques described can be extended to handle linear systems arising from other methods for two-point boundary value problems, such as multiple shooting and collocation. An application to seismic ray tracing is discussed.
Erdogan, Fazil
2008-02-01
In this lecture after some introductory remarks, first certain concepts relating to the mixed boundary value problems in mechanics of materials and in potential theory will be defined and typical examples from crack and contact mechanics demonstrating unique applications of the singular integral equations will be described. Then the methods of solution of the singular integral equations based mostly on the solution by orthogonal polynomials will very briefly be outlined and some future research particularly in the field of fracture mechanics will be discussed.
Positive Solutions of Singular Boundary Value Problem of Negative Exponent Emden–Fowler Equation
Indian Academy of Sciences (India)
Yuxia Wang; Xiyu Liu
2003-05-01
This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden–Fowler equation. A necessary and sufficient condition for the existence of [0, 1] positive solutions as well as 1[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.
The Graviton in the AdS-CFT correspondence Solution via the Dirichlet Boundary value problem
Mück, W
1998-01-01
Using the AdS-CFT correspondence we calculate the two point function of CFT energy momentum tensors. The AdS gravitons are considered by explicitly solving the Dirichlet boundary value problem for $x_0=\\epsilon$. We consider this treatment as complementary to existing work, with which we make contact.
Directory of Open Access Journals (Sweden)
Jian Liu
2013-09-01
Full Text Available In this article, we consider the free boundary value problem for one-dimensional compressible bipolar Navier-Stokes-Possion (BNSP equations with density-dependent viscosities. For general initial data with finite energy and the density connecting with vacuum continuously, we prove the global existence of the weak solution. This extends the previous results for compressible NS [27] to NSP.
On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem
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Ozgur Yildirim
2015-01-01
Full Text Available This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.
Solvability for a Class of Abstract Two-Point Boundary Value Problems Derived from Optimal Control
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Wang Lianwen
2007-01-01
Full Text Available The solvability for a class of abstract two-point boundary value problems derived from optimal control is discussed. By homotopy technique existence and uniqueness results are established under some monotonic conditions. Several examples are given to illustrate the application of the obtained results.
Solvability for a Class of Abstract Two-Point Boundary Value Problems Derived from Optimal Control
Directory of Open Access Journals (Sweden)
Lianwen Wang
2008-01-01
Full Text Available The solvability for a class of abstract two-point boundary value problems derived from optimal control is discussed. By homotopy technique existence and uniqueness results are established under some monotonic conditions. Several examples are given to illustrate the application of the obtained results.
Boundary Value Problems for First-Order Impulsive Functional q-Integrodifference Equations
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Jessada Tariboon
2014-01-01
Full Text Available We discuss the existence and uniqueness of solutions for a first-order boundary value problem for impulsive functional qk-integrodifference equations. The main results are obtained with the aid of some classical fixed point theorems. Illustrative examples are also presented.
Institute of Scientific and Technical Information of China (English)
徐西安
2004-01-01
In this paper, we first obtain some New results about the existence of multiple positive solutions for singular impulsive boundary value problems, and then to illustrate our main results we studied the existence of multiple positive solutions for an infinite system of scalar equations.
Boundary-value problems for first and second order functional differential inclusions
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Shihuang Hong
2003-03-01
Full Text Available This paper presents sufficient conditions for the existence of solutions to boundary-value problems of first and second order multi-valued differential equations in Banach spaces. Our results obtained using fixed point theorems, and lead to new existence principles.
Directory of Open Access Journals (Sweden)
Archana Chauhan
2012-12-01
Full Text Available In this article, we establish a general framework for finding solutions for impulsive fractional integral boundary-value problems. Then, we prove the existence and uniqueness of solutions by applying well known fixed point theorems. The obtained results are illustrated with an example for their feasibility.
NONTRIVIAL SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR FOURTH-ORDER DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
The singular boundary value problems for fourth-order differential equations are considered under some conditions concerning the first eigenvalues of the relevant linear operators. Sufficient conditions which guarantee the existence of nontrivial solutions are obtained. We use the topological degree to prove our main results.
THREE-POINT BOUNDARY VALUE PROBLEM FOR p-LAPLACIAN DIFFERENTIAL EQUATION AT RESONANCE
Institute of Scientific and Technical Information of China (English)
Minggang Zong; Wenyi Cai
2009-01-01
By topological degree theory, the three-point boundary value problem for p-Laplacian differential equation at resonance is studied. Some new results on the existence of so-lutions are obtained, which improve and extend some known ones in the previous literatures.
T, M P Ramirez
2012-01-01
Using a conjecture that allows to approach separable-variables conductivity functions, the elements of the Modern Pseudoanalytic Function Theory are used, for the first time, to numerically solve the Dirichlet boundary value problem of the two-dimensional Electrical Impedance Equation, when the conductivity function arises from geometrical figures, located within bounded domains.
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Hongxia Fan
2011-01-01
Full Text Available By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second-order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.
The Method of Subsuper Solutions for Weighted p(r-Laplacian Equation Boundary Value Problems
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Zhimei Qiu
2008-10-01
Full Text Available This paper investigates the existence of solutions for weighted p(r-Laplacian ordinary boundary value problems. Our method is based on Leray-Schauder degree. As an application, we give the existence of weak solutions for p(x-Laplacian partial differential equations.
Symbolic Iterative Solution of Boundary Value Problems for Partial Differential Equations
Semiyari, Hamid
2016-01-01
In this article we introduce a simple straightforward and powerful method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of partial differential boundary value problems. The method is easy to implement, computationally efficient, and it is highly accurate. The output of the method is a function that approximates the exact solution.
Institute of Scientific and Technical Information of China (English)
高永馨; 谢燕华
2009-01-01
利用上下解的方法研究了非线性2n阶常微分方程Y~((2n))=f(t,y,y',…,y~((2n-1))满足如下边界条件条件g_0(y(a),y'(a))=0,g_1(y'(a),y"(a),…,y~((2n-3))(a))=0,g_2(y~((2n-2))(a),y~((2n-1))(a))=0,h_0(y(c),y'(c),y"(c))=0,h_i(y~((i))(c),Y(i+1)(c))=0(i=3,4,…,2n-2).的非线性两点边值问题解的存在性.%By using the method of upper-lower solutions,the sufficient conditions are given for the existence of solutions to nonlinear two point boundary value problems for nonlinear 2nth-order differential equation y~((2n))=f(t,y,y',...,y~((2n-1))) with the boundary conditions g_0(y(a),y'(a)) =0,g_1(y'(a),y"(a),...,y~((2n-3))(a)) =0,g_2(y~((2n-2))(a),y~((2n-1))(a)) =0,h_0(y(c),y'(c),y"(c))=0,h_i(y~((i))(c),y~((i+1))(c))=0(i=3,4,...,2n-2).
Contact of boundary-value problems and nonlocal problems in mathematical models of heat transfer
Lyashenko, V.; Kobilskaya, O.
2015-10-01
In this paper the mathematical models in the form of nonlocal problems for the two-dimensional heat equation are considered. Relation of a nonlocal problem and a boundary value problem, which describe the same physical heating process, is investigated. These problems arise in the study of the temperature distribution during annealing of the movable wire and the strip by permanent or periodically operating internal and external heat sources. The first and the second nonlocal problems in the mobile area are considered. Stability and convergence of numerical algorithms for the solution of a nonlocal problem with piecewise monotone functions in the equations and boundary conditions are investigated. Piecewise monotone functions characterize the heat sources and heat transfer conditions at the boundaries of the area that is studied. Numerous experiments are conducted and temperature distributions are plotted under conditions of internal and external heat sources operation. These experiments confirm the effectiveness of attracting non-local terms to describe the thermal processes. Expediency of applying nonlocal problems containing nonlocal conditions - thermal balance conditions - to such models is shown. This allows you to define heat and mass transfer as the parameters of the process control, in particular heat source and concentration of the substance.
The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity
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Roman Janjgava
2016-01-01
Full Text Available We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems.
Directory of Open Access Journals (Sweden)
Hongwu Zhang
2011-08-01
Full Text Available In this article, we study a Cauchy problem for an elliptic equation with variable coefficients. It is well-known that such a problem is severely ill-posed; i.e., the solution does not depend continuously on the Cauchy data. We propose a modified quasi-boundary value regularization method to solve it. Convergence estimates are established under two a priori assumptions on the exact solution. A numerical example is given to illustrate our proposed method.
Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach
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Panos K. Palamides
2012-01-01
Full Text Available We investigate the existence of positive or a negative solution of several classes of four-point boundary-value problems for fourth-order ordinary differential equations. Although these problems do not always admit a (positive Green's function, the obtained solution is still of definite sign. Furthermore, we prove the existence of an entire continuum of solutions. Our technique relies on the continuum property (connectedness and compactness of the solutions funnel (Kneser's Theorem, combined with the corresponding vector field.
MULTIPLE POSITIVE SOLUTIONS TO A SINGULAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we study a singular third-order three-point boundary value problem. By a fixed point theorem of cone expansion-compression type due to Krasnosel'skii,we obtain various new results on the existence of two positive solutions to the problem,whose coefficient is allowed to have suitable singularities. Finally,we give an example to verify our results.
SINGULARLY PERTURBED SEMI-LINEAR BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS FUNCTION
Institute of Scientific and Technical Information of China (English)
Ding Haiyun; Ni Mingkang; Lin Wuzhong; Cao Yang
2012-01-01
A class of singularly perturbed semi-linear boundary value problems with discontinuous functions is examined in this article.Using the boundary layer function method,the asymptotic solution of such a problem is given and shown to be uniformly effective. The existence and uniqueness of the solution for the system is also proved.Numerical result is presented as an illustration to the theoretical result.
Generalized solutions of initial-boundary value problems for second-order hyperbolic systems
Alexeyeva, L. A.; Zakir'yanova, G. K.
2011-07-01
The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green's matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.
On explicit and numerical solvability of parabolic initial-boundary value problems
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Lepsky Olga
2006-01-01
Full Text Available A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain ( being a bounded domain, and being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution . In the paper, the result is obtained not just for the operator , but also for an arbitrary parabolic differential operator , where is an elliptic operator in of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation in is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables.
The initial boundary value problem for free-evolution formulations of General Relativity
Hilditch, David
2016-01-01
We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate primarily on boundaries that are geometrically determined by the outermost normal observer to spacelike slices of the foliation. We present high-order-derivative boundary conditions for the gauge, constraint violating and gravitational wave degrees of freedom of the formulation. Second order derivative boundary conditions are presented in terms of the conformal variables used in numerical relativity simulations. Using Kreiss-Agranovich-Metivier theory we demonstrate, in the frozen coefficient approximation, that with sufficiently high order derivative boundary conditions the initial boundary value problem can be rendered boundary stable. The precise number of derivatives required depends on the gauge. For a choice of the gauge condition that renders the system strongly hyperbolic...
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Jeffrey W. Lyons
2017-01-01
Full Text Available For \\(\\alpha\\in(1,2]\\, the singular fractional boundary value problem \\[D^{\\alpha}_{0^+}x+f\\left(t,x,D^{\\mu}_{0^+}x\\right=0,\\quad 0\\lt t\\lt 1,\\] satisfying the boundary conditions \\(x(0=D^{\\beta}_{0^+}x(1=0\\, where \\(\\beta\\in(0,\\alpha-1]\\, \\(\\mu\\in(0,\\alpha-1]\\, and \\(D^{\\alpha}_{0^+}\\, \\(D^{\\beta}_{0^+}\\ and \\(D^{\\mu}_{0^+}\\ are Riemann-Liouville derivatives of order \\(\\alpha\\, \\(\\beta\\ and \\(\\mu\\ respectively, is considered. Here \\(f\\ satisfies a local Carathéodory condition, and \\(f(t,x,y\\ may be singular at the value 0 in its space variable \\(x\\. Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.
Institute of Scientific and Technical Information of China (English)
袁益让
1996-01-01
The software for oil-gas transport and accumulation is to describe the history of oil-gas transport and accumulation in basin evolution. It is of great value in rational evaluation of prospecting and exploiting oil-gas resources. This thesis, from actual conditions such as the effects of gravitation, buoyancy and capillary pressure, puts forward for the two class boundary value problem a kind of characteristic mixed finite element scheme by making use of the change of region, time step modified techniques of handling boundary value condition, negative norm estimate and the theory of prior estimates. Optimal order estimates in L2 norm are derived for the error in approximate solutions. Thus the well-known theoretical problem proposed by J. Douglas, Jr has been thoroughly and completely solved.
Monotone methods for solving a boundary value problem of second order discrete system
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Wang Yuan-Ming
1999-01-01
Full Text Available A new concept of a pair of upper and lower solutions is introduced for a boundary value problem of second order discrete system. A comparison result is given. An existence theorem for a solution is established in terms of upper and lower solutions. A monotone iterative scheme is proposed, and the monotone convergence rate of the iteration is compared and analyzed. The numerical results are given.
2007-01-01
In the present work, we study a multi-point boundary-value problem for an ordinary differential equation with matrix coefficients containing a spectral parameter in the boundary conditions. Assuming some regularity conditions, we show that the characteristic determinant has an infinite number of zeros, and specify their asymptotic behavior. Using the asymptotic behavior of Green matrix on contours expending at infinity, we establish the series expansion formula of sufficiently smooth function...
IMPULSIVE BOUNDARY VALUE PROBLEMS FOR STURM-LIOUVILLE TYPE DIFFERENTIAL INCLUSIONS
Institute of Scientific and Technical Information of China (English)
Yicheng LIU; Jun WU; Zhixiang LI
2007-01-01
In this paper, the authors investigate the existence of solutions of impulsive boundary value problems for Sturm-Liouville type differential inclusions which admit non-convex-valued multifunctions on right hand side. Two results under weaker conditions are presented. The methods rely on a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler and Schaefer's fixed point theorem combined with lower semi-continuous multi-valued operators with decomposable values.
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Domoshnitsky Alexander
2009-01-01
Full Text Available We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
GLOBAL SOLUTIONS TO AN INITIAL BOUNDARY VALUE PROBLEM FOR THE MULLINS EQUATION
Institute of Scientific and Technical Information of China (English)
Hans-Dieter Alber; Zhu Peicheng
2007-01-01
In this article we study the global existence of solutions to an initial boundary value problem for the Mullins equation which describes the groove development at the grain boundaries of a heated polycrystal, both the Dirichlet and the Neumann boundary conditions are considered. For the classical solution we also investigate the large time behavior, it is proved that the solution converges to a constant, in the L∞(Ω)-norm, as time tends to infinity.
Infinitely many solutions to superlinear second order
Chen Xiaoqiang; Ma Ruyun; Gao Chenghua
2011-01-01
Abstract We consider the boundary value problem u ″ ( x ) + g ( u ( x ) ) + p ( x , u ( x ) , u ′ ( x ) ) = 0 , x ∈ ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m - 2 α i u ( η i ) , where: (1) m ≥ 3, ηi ∈ (0, 1) and αi > 0 with A : = ∑ i = 1 m - 2 α i < 1 ; (2) g : ℝ → ℝ is continuous and sat...
Infinitely many solutions for a fourth-order boundary-value problem
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Seyyed Mohsen Khalkhali
2012-09-01
Full Text Available In this article we consider the existence of infinitely many solutions to the fourth-order boundary-value problem $$displaylines{ u^{iv}+alpha u''+eta(x u=lambda f(x,u+h(u,quad xin]0,1[cr u(0=u(1=0,cr u''(0=u''(1=0,. }$$ Our approach is based on variational methods and critical point theory.
Mishra, Siddhartha
2011-01-01
Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physically relevant solution. Numerical experiments illustrating the robust performance of these schemes are presented.
Existence of Sign-changing Solution for Three-point Boundary Value Problems
Institute of Scientific and Technical Information of China (English)
LI Chun-yan; SU Ya-juan
2012-01-01
In this paper,by using the fixed-point index theory,we study the existence of sign-changing solution of some three-point boundary value problems {y″(t) + f(y) =0, t ∈ [0,1],y′(0) =0, y( 1 ) =αy(η),where 0 ＜ a ＜ 1,0 ＜ η ＜ 1,f:R → R is continuous,strictly increasing and f(0) =0.
Solution of the boundary value problem for optimal escape in continuous stochastic systems and maps.
S; Beri; Mannella, R.; Luchinsky, Dmitry G.; Silchenko, A. N.; McClintock, Peter V. E.
2005-01-01
Topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps. A topological method is introduced that simplifies the solution of boundary value problems: The activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path. The method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps.
Oaku, Toshinori
1986-01-01
We give a general formulation of boundary value problems in the framework of hyperfunctions both for systems of linear partial differential equations with non-characteristic boundary and for Fuchsian systems of partial differential equations in a unified manner. We also give a microlocal formulation, which enables us to prove new results on propagation of micro-analyticity up to the boundary for solutions of systems micro-hyperbolic in a weak sense.
Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method
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Mustafa Inc
2014-01-01
Full Text Available We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM, homotopy perturbation method (HPM, Adomian decomposition method (ADM, variation of parameters method (VPM, and homotopy analysis method (HAM. Obtained results show that our method is very effective.
A Simulation Study of the Overdetermined Geodetic Boundary Value Problem Using Collocation
1989-03-01
boundary by means of spherical harmonics ( Heiskanen and Moritz, 1967). The third or mixed boundary value problem of potential theory is the determination...together with the necessity of eliminating the masses exterior to the ellipsoid, ( Heiskanen and Moritz, 1967; Moritz, 1980). In this manner the...functions. After performing the integration in equation (2.2-15) ( Heiskanen and Moritz, 1967, pp. 257-258) and considering no zero and first degree
VAGO method for the solution of elliptic second-order boundary value problems
Vabishchevich, Nikolay P
2010-01-01
Mathematical physics problems are often formulated using differential oprators of vector analysis - invariant operators of first order, namely, divergence, gradient and rotor operators. In approximate solution of such problems it is natural to employ similar operator formulations for grid problems, too. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection-diffusion-reaction equation the diffusion coefficient is a symmetric tensor of second order.
Student Solutions Manual to Boundary Value Problems and Partial Differential Equations
Powers, David L
2005-01-01
This student solutions manual accompanies the text, Boundary Value Problems and Partial Differential Equations, 5e. The SSM is available in print via PDF or electronically, and provides the student with the detailed solutions of the odd-numbered problems contained throughout the book.Provides students with exercises that skillfully illustrate the techniques used in the text to solve science and engineering problemsNearly 900 exercises ranging in difficulty from basic drills to advanced problem-solving exercisesMany exercises based on current engineering applications
Institute of Scientific and Technical Information of China (English)
YUAN Yi-rang; LI Chang-feng; YANG Cheng-shun; HAN Yu-ji
2008-01-01
The coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution. It is of great value in rational evaluation of prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. A kind of characteristic finite difference schemes is put forward, from which optimal order estimates in l2 norm are derived for the error in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method and software development.
Zhu, Changjiang; Duan, Renjun
2003-02-01
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation \\left\\{\\begin{array}{@{}l@{\\qquad}l@{}} u_t+\\big(\\frac{u^2}{2}\\big)_x=0 x\\gt0\\quad t\\gt0\\\\ u(x,0)=u_0(x) x\\geq0\\\\ u(0,t)=0 t\\geq0. \\end{array}\\right. To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in Evans (1998 Partial Differential Equations (Providence, RI: American Mathematical Society) and Hopf (1950 Commun. Pure Appl. Math. 3 201-30), give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation.
PERIODIC BOUNDARY VALUE PROBLEM OF QUASILINEAR SYSTEM%拟线性系统周期边值问题
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The existence and uniqueness results about quasilinear periodic boundary value problems are established by using the global inverse function theorem and the result about the existence and uniquencess of periodic solutions for the periodic boundary value problem of nonhomogeneous linear periodic system.
EXISTENCE OF SOLUTIONS TO AN m-POINT BOUNDARY VALUE PROBLEM OF THIRD ORDER ODE’S AT RESONANCE
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we study an m-point boundary value problem of third order ODEs at resonance. We prove some existence results for the m-point boundary value problem at resonance by the coincidence degree theory of [8,9]. Our result is new. Meanwhile,an example is presented to demonstrate the main result.
Institute of Scientific and Technical Information of China (English)
沈建和; 周哲彦; 余赞平
2009-01-01
In this paper,existence of solutions of third-order differential equationy (t) = f(t,y(t),y'(t),y"(t))with nonlinear three-point boundary condition g(y(a),y'(a),y"(a)) = 0,h(y(b),y'(b))=0,I(y(c),y'(c),y"(c)) = 0is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a,b,c ∈ R,a < b< c; f:[a,c] × R3a → R,g:R3 → R,h:R2→R and I:R3 → Rare continuous functions.The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem.As an application,an example with an explicit solution is given to demonstrate the validity of the results in this paper.
Institute of Scientific and Technical Information of China (English)
高永馨; 谢燕华
2012-01-01
利用上下解方法,讨论了四阶微分方程非线性两点边值问题y(4) =f(x,y,y′,y″,y(′″)),y(b) =b0,y′(b) =b1,y″(b) =h(y″(a)),g(y(a),y(b),y′(a),y′(b),y″(a),y″(b),y(′″)(a),y(′″)(b)) =0解的存在唯一性.%By using the method of upper - lower solution,the existence and uniquenss of solutions of nonlinear two -point boundary value problems for fourth order differential equation y(4) =f(x,y,y′,y″,y(′″)),y(b) =b0,y′(b) =b1,y″(b) =h(y″(a)),g(y(a),y(b),y′(a),y′(b),y″(a),y″(b),y(′″)(a),y(′″)(b)) =0 are investigated.
Directory of Open Access Journals (Sweden)
George N. Galanis
2005-10-01
Full Text Available In this paper we prove the existence of positive solutions for the three-point singular boundary-value problem$$ -[phi _{p}(u']'=q(tf(t,u(t,quad 0
Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations
Directory of Open Access Journals (Sweden)
Olivier Sarbach
2012-08-01
Full Text Available Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
Directory of Open Access Journals (Sweden)
Mohamed Denche
2007-01-01
Full Text Available In the present work, we study a multi-point boundary-value problem for an ordinary differential equation with matrix coefficients containing a spectral parameter in the boundary conditions. Assuming some regularity conditions, we show that the characteristic determinant has an infinite number of zeros, and specify their asymptotic behavior. Using the asymptotic behavior of Green matrix on contours expending at infinity, we establish the series expansion formula of sufficiently smooth functions in terms of residuals solutions to the given problem. This formula actually gives the completeness of root functions as well as the possibility of calculating the coefficients of the series.
Solution of Seventh Order Boundary Value Problems by Variation of Parameters Method
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Muzammal Iftikhar
2013-01-01
Full Text Available The induction motor behavior is represented by a fifth order differential equation model. Addition of a torque correction factor to this model accurately reproduces the transient torques and instantaneous real and reactive power flows of the full seventh order differential equation model. The aim of this study is to solve the seventh order boundary value problems and the variation of parameters method is used for this purpose. The approximate solutions of the problems are obtained in terms of rapidly convergent series. Two numerical examples have been given to illustrate the efficiency and implementation of the method.
OpenMP for 3D potential boundary value problems solved by PIES
KuŻelewski, Andrzej; Zieniuk, Eugeniusz
2016-06-01
The main purpose of this paper is examination of an application of modern parallel computing technique OpenMP to speed up the calculation in the numerical solution of parametric integral equations systems (PIES). The authors noticed, that solving more complex boundary problems by PIES sometimes requires large computing time. This paper presents the use of OpenMP and fast C++ linear algebra library Armadillo for boundary value problems modelled by 3D Laplace's equation and solved using PIES. The testing example shows that the use of mentioned technologies significantly increases speed of calculations in PIES.
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Muhammad Aslam Noor
2008-01-01
Full Text Available We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. This method is called the variational homotopy perturbation method (VHPM. We use this method for solving higher dimensional initial boundary value problems with variable coefficients. The developed algorithm is quite efficient and is practically well suited for use in these problems. The proposed scheme finds the solution without any discritization, transformation, or restrictive assumptions and avoids the round-off errors. Several examples are given to check the reliability and efficiency of the proposed technique.
Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons
Institute of Scientific and Technical Information of China (English)
Hongli JIA; Benyu GUO
2010-01-01
The authors investigate Petrov-Galerkin spectral element method.Some results on Legendre irrational quasi-orthogonal approximations are established,which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons.As examples of applications,spectral element methods for two model problems,with the spectral accuracy in certain Jacobi weighted Sobolev spaces,are proposed.The techniques developed in this paper are also applicable to other higher order methods.
Institute of Scientific and Technical Information of China (English)
黄民海
2011-01-01
A spectral transform method for solving initial boundary-value problems for linear and for integral nonlinear PDEs has been introduced recently by Fokas.By using this method,the mixed boundary value problem for the modified Helmholtz equation in a wedge is discussed.The problem is changed to solve a Riemann-Hilbert boundary value problem.Finally,the integral representation of the solution is obtained in closed form.%考虑一类楔形域上Modified-Helmholtz方程的混合边值问题.利用新型的Fokas谱变换方法,将问题转化为求解一类Riemann-Hilbert边值问题,从而得到了方程解的封闭积分表达式.
Second-order schemes for a boundary value problem with Neumann's boundary conditions
Dehghan, Mehdi
2002-01-01
A new second-order finite difference scheme based on the (3, 3) alternating direction implicit method and a new second-order finite difference technique based on the (5, 5) implicit formula are discussed for solving a nonlocal boundary value problem for the two-dimensional diffusion equation with Neumann's boundary conditions. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of the one-dimensional case. Using a suitable transformation the solution of this problem is equivalent to the solution of two other problems. The former, which is a one-dimensional nonlocal boundary value problem giving the value of [mu] through using the unconditionally stable standard implicit (3, 1) backward time-centred space (denoted BTCS) scheme. Using this result the second problem will be changed to a classical two-dimensional diffusion equation with Neumann's boundary conditions which will be solved numerically by using the unconditionally stable alternating direction implicit (3, 3) technique or the fully implicit finite difference scheme. The results of a numerical example are given and computation times are presented. Error estimates derived in the maximum norm are also tabulated.
Boundary-value problems for wave equations with data on the whole boundary
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Makhmud A. Sadybekov
2016-10-01
Full Text Available In this article we propose a new formulation of boundary-value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of boundary-value problem in the classical and generalized senses. To substantiate the well-posedness of this problem it is necessary to have an effective representation of the general solution of the problem. In this direction we obtain a convenient representation of the general solution for the wave equation in a rectangular domain based on d'Alembert classical formula. The constructed general solution automatically satisfies the boundary conditions by a spatial variable. Further, by setting different boundary conditions according to temporary variable, we get some functional or functional-differential equations. Thus, the proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations.
Solving Directly Two Point Non Linear Boundary Value Problems Using Direct Adams Moulton Method
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Zanariah A. Majid
2011-01-01
Full Text Available Problem statement: In this study, a direct method of Adams Moulton type was developed for solving non linear two point Boundary Value Problems (BVPs directly. Most of the existence researches involving BVPs will reduced the problem to a system of first order Ordinary Differential Equations (ODEs. This approach is very well established but it obviously will enlarge the systems of first order equations. However, the direct method in this research will solved the second order BVPs directly without reducing it to first order ODEs. Approach: Lagrange interpolation polynomial was applied in the derivation of the proposed method. The method was implemented using constant step size via shooting technique in order to determine the approximated solutions. The shooting technique will employ the Newtons method for checking the convergent of the guessing values for the next iteration. Results: Numerical results confirmed that the direct method gave better accuracy and converged faster compared to the existing method. Conclusion: The proposed direct method is suitable for solving two point non linear boundary value problems.
A simple finite element method for boundary value problems with a Riemann–Liouville derivative
Jin, Bangti
2016-02-01
© 2015 Elsevier B.V. All rights reserved. We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα-^{1} in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and ^{L2}(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.
General theory of spherically symmetric boundary-value problems of the linear transport theory.
Kanal, M.
1972-01-01
A general theory of spherically symmetric boundary-value problems of the one-speed neutron transport theory is presented. The formulation is also applicable to the 'gray' problems of radiative transfer. The Green's function for the purely absorbing medium is utilized in obtaining the normal mode expansion of the angular densities for both interior and exterior problems. As the integral equations for unknown coefficients are regular, a general class of reduction operators is introduced to reduce such regular integral equations to singular ones with a Cauchy-type kernel. Such operators then permit one to solve the singular integral equations by the standard techniques due to Muskhelishvili. We discuss several spherically symmetric problems. However, the treatment is kept sufficiently general to deal with problems lacking azimuthal symmetry. In particular the procedure seems to work for regions whose boundary coincides with one of the coordinate surfaces for which the Helmholtz equation is separable.
Free boundary value problem to 3D spherically symmetric compressible Navier-Stokes-Poisson equations
Kong, Huihui; Li, Hai-Liang
2017-02-01
In the paper, we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier-Stokes-Poisson equations for self-gravitating gaseous stars with γ -law pressure density function for 6/5 <γ ≤ 4/3. For stress-free boundary condition and zero flow density continuously across the free boundary, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated for spherically symmetric initial data with the total mass smaller than a critical mass.
Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
DEFF Research Database (Denmark)
Abels, Helmut; Grubb, Gerd; Wood, Ian Geoffrey
2014-01-01
The theory of selfadjoint extensions of symmetric operators, and more generally the theory of extensions of dual pairs, was implemented some years ago for boundary value problems for elliptic operators on smooth bounded domains. Recently, the questions have been taken up again for nonsmooth domains....... In the present work we show that pseudodifferential methods can be used to obtain a full characterization, including Kreĭn resolvent formulas, of the realizations of nonselfadjoint second-order operators on
A MIXED ELECTRIC BOUNDARY VALUE PROBLEM FOR AN ANTI－PLANE PIEZOELECTRIC CRACK
Institute of Scientific and Technical Information of China (English)
ttnAngZlaenyu; KuangZhenbang
2003-01-01
The analytical continuation method is adopted to solve a mixed electric boundary value problem for a piezoelectric medium under anti-plane deformation. The crack face is partly conductive and partly impermeable. The results show that the stress intensity factor is identical with the mode III stress intensity factor independent of the conducting length. But the electric field and the electric displacement are dependent on the electric boundary conditions on the crack faces and are singular not only at the crack tips but also at the junctures between the impermeable part and conducting portions.
Thanin Sitthiwirattham; Jessada Tariboon
2012-01-01
By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem ${\\Delta }^{2}u(t-1)+a(t)f(u(t))=0$ , $t\\in \\{1,2,\\dots ,T\\}$ , $u(0)=\\beta {\\sum }_{s=1}^{\\eta }u(s)$ , $u(T+1)=\\alpha {\\sum }_{s=1}^{\\eta }u(s)$ , where $f$ is continuous, $T\\ge 3$ is a fixed positive integer, $\\eta \\in \\{1,2,...,T-1\\}$ , $0
A NEW EFFICIENT METHOD TO BOUNDARY VALUE PROBLEM FOR BALLISTIC ROCKET GUIDANCE
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The exploitation of rocket guidance technology on the basis of the guidance law of Space Shuttle and Pegasus rocket was performed. A new efficient method of numerical iteration solution to the boundary value problem was put forward. The numerical simulation results have shown that the method features good performances of stability, robustness, high precision, and algebraic formulas in real computation. By virtue of modern DSP (digital signal processor) high speed chip technology, the algorithm can be used in real time and can adapt to the requirements of the big primary bias of rocket guidance.
Boundary Value Problems for Singular Second-Order Functional Differential Equations
Institute of Scientific and Technical Information of China (English)
Hui-zhao Liu; Da-qing Jiang; Yan Wang
2002-01-01
Positive solutions to the boundary value problem,{y"=-f(x,y(w(x))),0〈x〈1,αy(x)-βy′(x)=ξ(x),a≤x≤0,γy(x)+δy′(x)=η(x),1≤x≤b, are obtained by applying the Schauder fixed point theorem, where w(x) is a continuous function defined on [0,1] and f(x,y) is a function defined on (0,1)× (0,∞), which satisfies certain restrictions and may have singularity at y=0. The result corrects and improves an existence theorem due to Erbe and Kong[1].
Building Infinitely Many Solutions for Some Model of Sublinear Multipoint Boundary Value Problems
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Guy Aymard Degla
2015-01-01
Full Text Available We show that the sublinearity hypothesis of some well-known existence results on multipoint Boundary Value Problems (in short BVPs may allow the existence of infinitely many solutions by using Tietze extension theorem. This is a qualitative result which is of concern in Applied Analysis and can motivate more research on the conditions that ascertain the existence of multiple solutions to sublinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differential equations with multiple solutions.
The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in Cn
Institute of Scientific and Technical Information of China (English)
Liang Yu LIN; Chun Hui QIU
2004-01-01
Suppose that D is a bounded domain with a piecewise C1 smooth boundary in Cn. Let ψ∈ C1+α((б)D). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Bochner-Martinelli kernel, which has integral density ψ. Moreover,by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauchy boundary value problem with the Bochner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.
Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Weak solution (or generalized solution) for the boundary-value problems of partial differential equations of elasticity of 3D (three-dimensional) quasicrystals is given,in which the matrix expression is used. In terms of Korn inequality and theory of function space, we prove the uniqueness of the weak solution. This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals, and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students.
A combined analytic-numeric approach for some boundary-value problems
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Mustafa Turkyilmazoglu
2016-02-01
Full Text Available A combined analytic-numeric approach is undertaken in the present work for the solution of boundary-value problems in the finite or semi-infinite domains. Equations to be treated arise specifically from the boundary layer analysis of some two and three-dimensional flows in fluid mechanics. The purpose is to find quick but accurate enough solutions. Taylor expansions at either boundary conditions are computed which are next matched to the other asymptotic or exact boundary conditions. The technique is applied to the well-known Blasius as well as Karman flows. Solutions obtained in terms of series compare favorably with the existing ones in the literature.
Numerical study of a parametric parabolic equation and a related inverse boundary value problem
Mustonen, Lauri
2016-10-01
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the nonhomogeneous diffusion coefficient in the interior of an object. The method in this paper relies on solving the forward problem for a whole family of diffusivities by using a spectral Galerkin method in the high-dimensional parameter domain. The evaluation of the parametric solution and its derivatives is then completely independent of spatial and temporal discretizations. In the case of a quadratic approximation for the parameter dependence and a direct solver for linear least squares problems, we show that the evaluation of the parametric solution does not increase the complexity of any linearized subproblem arising from a Gauss-Newtonian method that is used to minimize a Tikhonov functional. The feasibility of the proposed algorithm is demonstrated by diffusivity reconstructions in two and three spatial dimensions.
Conjugate Gradient Method with Ritz Method for the Solution of Boundary Value Problems
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Victor Onomza WAZIRI
2007-01-01
Full Text Available In this paper, we wish to determine the optimal control of a one-variable boundary value problem using the Ritz algorithm. The posed optimal control problem was inadequate to achieve our goal using the Conjugate Gradient Method version developed by (Harsdoff, 1976. It is anticipated that other operators from some given different problems may sustain the application of the algorithm if the approximate solutions terms are properly chosen quadratic functionals. The graphical solution given at the end of section five of the paper, however, shows that our problem can not have an optimal minimum value since the minimum output is not unique. The optimal value obtained using Mathcad program codes may constitute a conjugate gradient approximate numerical value. As observed from the graphical output, Ritz algorithm could give credence for wider horizon in the engineering computational methods for vibrations of mechanical components and simulates.
THE UPWIND FINITE DIFFERENCE METHOD FOR MOVING BOUNDARY VALUE PROBLEM OF COUPLED SYSTEM
Institute of Scientific and Technical Information of China (English)
Yuan Yirang
2011-01-01
Coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution. It is of great value in rational evaluation of prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. The upwind finite difference schemes applicable to parallel arith- metic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as change of variables, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order dif- ference operators and prior estimates, are adopted. The estimates in 12 norm are derived to determine the error in the approximate solution. This method was already applied to the numerical simulation of migration-accumulation of oil resources.
Solvability of fractional multi-point boundary-value problems with p-Laplacian operator at resonance
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Tengfei Shen
2014-02-01
Full Text Available In this article, we consider the multi-point boundary-value problem for nonlinear fractional differential equations with $p$-Laplacian operator: $$\\displaylines{ D_{0^+}^\\beta \\varphi_p (D_{0^+}^\\alpha u(t = f(t,u(t,D_{0^+}^{\\alpha - 2} u(t,D_{0^+}^{\\alpha - 1} u(t, D_{0^+}^\\alpha u(t,\\quad t \\in (0,1, \\cr u(0 = u'(0=D_{0^+}^\\alpha u(0 = 0,\\quad D_{0^+}^{\\alpha - 1} u(1 = \\sum_{i = 1}^m {\\sigma_i D_{0^+}^{\\alpha - 1} u(\\eta_i } , }$$ where $2 < \\alpha \\le 3$, $0 < \\beta \\le 1$, $3 < \\alpha + \\beta \\le 4$, $\\sum_{i = 1}^m {\\sigma_i } = 1$, $D_{0^+}^\\alpha$ is the standard Riemann-Liouville fractional derivative. $\\varphi_{p}(s=|s|^{p-2}s$ is p-Laplacians operator. The existence of solutions for above fractional boundary value problem is obtained by using the extension of Mawhin's continuation theorem due to Ge, which enrich konwn results. An example is given to illustrate the main result.
Khairullin, Ermek
2016-08-01
In this paper we consider a special boundary value problem for multidimensional parabolic integro-differential equation with boundary conditions that contains as a boundary condition containing derivatives of order higher than the order of the equation. The solution is sought in the form of a thermal potential of a double layer. Shows lemma of finding the limits of the derivatives of the unknown function in the neighborhood of the hyperplane. Using the boundary condition and lemma obtained integral-differential equation (IDE) of parabolic operators, whĐţre an unknown function under the integral contains higher-order space variables derivatives. IDE is reduced to a singular integral equation (SIE), when an unknown function in the spatial variables satisfies the Holder. The characteristic part is solved in the class of distribution function using method of transformation of Fourier-Laplace. Found an algebraic condition for the transition to the classical generalized solution. Integral equation of the resolvent for the characteristic part of SIE is obtained. Integro-differential equation is reduced to the Volterra-Fredholm type integral equation of the second kind by method of regularization. It is shown that the solution of SIE is a solution of IDE. Obtain a theorem on the solvability of the boundary value problem of multidimensional parabolic integro-differential equation, when a known function of the spatial variables belongs to the Holder class and satisfies the solvability conditions.
Variational solution about over-determined geodetic boundary value problem and its related theories
Institute of Scientific and Technical Information of China (English)
2007-01-01
A new solving method for Laplace equation with over-determined geodetic boundary conditions is pro- posed in the paper, with the help of minimizing some kinds of quadratic functional in calculus of variation. At first, the so-called variational solution for over-determined geodetic boundary value problem is defined in terms of principles in calculus of variation. Then theoretical properties related with the solution are derived, especially for its existence, uniqueness and optimal approximation. And then the computational method of the solution is discussed, and its expression is exhibited under the case that all boundaries are spheres. Finally an arithmetic example about EGM96 gravity field model is given, and the computational results show that the proposed method can efficiently raise accuracy to deal with gravity data. In all, the variational solution of over-determined geodetic boundary value problem can not only fit to deal with many kinds of gravity data in a united form, but also has strict mathematical basements.
Energy Technology Data Exchange (ETDEWEB)
Reinhardt, Hans-Juergen, E-mail: reinhardt@mathematik.uni-siegen.de [Department of Mathematics, University of Siegen, Emmy-Noether-Campus, Walter-Flex-Str. 3, D-57072 Siegen (Germany)
2011-04-01
In this paper singularly perturbed parabolic initial-boundary value problems are considered which, in addition, are illposed. The latter means that at one end of the 1-d spatial domain two conditions (for the solution and its spatial derivative) are given while on the other end the corresponding quantities are to be determined. It is well-known that such problems are illposed in the mathematical sense. Here, in addition, boundary layers may occur which make the problems more difficult. For relatively simple examples numerical experiments have been carried out and numerical results are shown. The Conjugate Gradient Methods is used to find the desired quantities iteratively. It will be explained what has to be done in any iteration step. A regularisation is performed by means of discretization and by determining an optimal final iteration step via a stopping rule.
Boundary value problem for the solution of magnetic cutoff rigidities and some special applications
Edmonds, Larry
1987-01-01
Since a planet's magnetic field can sometimes provide a spacecraft with some protection against cosmic ray and solar flare particles, it is important to be able to quantify this protection. This is done by calculating cutoff rigidities. An alternate to the conventional method (particle trajectory tracing) is introduced, which is to treat the problem as a boundary value problem. In this approach trajectory tracing is only needed to supply boundary conditions. In some special cases, trajectory tracing is not needed at all because the problem can be solved analytically. A differential equation governing cutoff rigidities is derived for static magnetic fields. The presense of solid objects, which can block a trajectory and other force fields are not included. A few qualititative comments, on existence and uniqueness of solutions, are made which may be useful when deciding how the boundary conditions should be set up. Also included are topics on axially symmetric fields.
Mixed Initial-Boundary Value Problem for Telegraph Equation in Domain with Variable Borders
Directory of Open Access Journals (Sweden)
V. A. Ostapenko
2012-01-01
Full Text Available Mixed initial-boundary value problem for telegraph equation in domain with variable borders is considered. On one part of domain’s border are the boundary conditions of the first type, on other part of the boundary are set boundary conditions of the second type. Besides, the sizes of area are variable. The solution of such problem demands development of special methods. With the help of consecutive application of procedure of construction waves reflected from borders of domain, it is possible to obtain the solution of this problem in quadratures. In addition, for construction of the waves reflected from mobile border, it is necessary to apply the procedure specially developed for these purposes.
The second boundary value problem for equations of viscoelastic diffusion in polymers
Vorotnikov, Dmitry A
2009-01-01
The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors. Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In this chapter we study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. We establish existence of weak solutions to this problem. We suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as tim...
On existence and uniqueness of positive solutions to a class of fractional boundary value problems
Directory of Open Access Journals (Sweden)
Caballero J
2011-01-01
Full Text Available Abstract The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem D 0 + α u ( t + f ( t , u ( t = 0 , 0 < t < 1 , u ( 0 = u ( 1 = u ′ ( 0 = 0 , where 2 < α ≤ 3 and D 0 + α is the Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear, the question of uniqueness of the solution is not treated. We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear. 2010 Mathematics Subject Classification: 34B15
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
By different fixed point theorems in cones, sufficient conditions for the existence and multiple existence of positive solutions to a class of second-order multi-point boundary value problem for dynamic equation on time scales are obtained.
Directory of Open Access Journals (Sweden)
Mabrouk Briki
2016-05-01
Full Text Available In this paper, a fourth-order boundary value problem on the half-line is considered and existence of solutions is proved using a minimization principle and the mountain pass theorem.
Directory of Open Access Journals (Sweden)
Muhammad Tahir
2012-11-01
Full Text Available This article shows the uniqueness of a solution to a Bitsadze system of equations, with a boundary-value problem that has four additional single point conditions. It also shows how to construct the solution.
Directory of Open Access Journals (Sweden)
Randhir Singh
2012-01-01
Full Text Available We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results.
Multiple positive solutions for fourth-order three-point p-Laplacian boundary-value problems
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Hanying Feng
2007-02-01
Full Text Available In this paper, we study the three-point boundary-value problem for a fourth-order one-dimensional $p$-Laplacian differential equation $$ ig(phi_p(u''(tig''+ a(tfig(u(tig=0, quad tin (0,1, $$ subject to the nonlinear boundary conditions: $$displaylines{ u(0=xi u(1,quad u'(1=eta u'(0,cr (phi _{p}(u''(0' =alpha _{1}(phi _{p}(u''(delta', quad u''(1=sqrt[p-1]{eta _{1}}u''(delta, }$$ where $phi_{p}(s=|s|^{p-2}s$, $p>1$. Using the five functional fixed point theorem due to Avery, we obtain sufficient conditions for the existence of at least three positive solutions.
The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line
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Xu, Jian, E-mail: jianxu@usst.edu.cn [University of Shanghai for Science and Technology, College of Science (China); Fan, Engui, E-mail: faneg@fudan.edu.cn [Fudan University, School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Science (China)
2016-09-15
We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a vector Riemann-Hilbert problem formulated in the plane of a complex spectral parameter z.
On explicit and numerical solvability of parabolic initial-boundary value problems
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Olga Lepsky
2006-05-01
Full Text Available A homogeneous boundary condition is constructed for the parabolic equation (Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”u=f in an arbitrary cylindrical domain ÃŽÂ©ÃƒÂ—Ã¢Â„Â (ÃŽÂ©Ã¢ÂŠÂ‚Ã¢Â„Ân being a bounded domain, I and ÃŽÂ” being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”, but also for an arbitrary parabolic differential operator Ã¢ÂˆÂ‚t+A, where A is an elliptic operator in Ã¢Â„Ân of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation (Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”u=0 in ÃŽÂ©ÃƒÂ—Ã¢Â„Â is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables.
METHOD OF GREEN FUNCTIONS IN MATHEMATICAL MODELLING FOR TWO-POINT BOUNDARY-VALUE PROBLEMS
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E. V. Dikareva
2015-01-01
Full Text Available Summary. In many applied problems of control, optimization, system theory, theoretical and construction mechanics, for problems with strings and nods structures, oscillation theory, theory of elasticity and plasticity, mechanical problems connected with fracture dynamics and shock waves, the main instrument for study these problems is a theory of high order ordinary differential equations. This methodology is also applied for studying mathematical models in graph theory with different partitioning based on differential equations. Such equations are used for theoretical foundation of mathematical models but also for constructing numerical methods and computer algorithms. These models are studied with use of Green function method. In the paper first necessary theoretical information is included on Green function method for multi point boundary-value problems. The main equation is discussed, notions of multi-point boundary conditions, boundary functionals, degenerate and non-degenerate problems, fundamental matrix of solutions are introduced. In the main part the problem to study is formulated in terms of shocks and deformations in boundary conditions. After that the main results are formulated. In theorem 1 conditions for existence and uniqueness of solutions are proved. In theorem 2 conditions are proved for strict positivity and equal measureness for a pair of solutions. In theorem 3 existence and estimates are proved for the least eigenvalue, spectral properties and positivity of eigenfunctions. In theorem 4 the weighted positivity is proved for the Green function. Some possible applications are considered for a signal theory and transmutation operators.
Hasni, Mohd Mughti; Majid, Zanariah Abdul; Senu, Norazak; Rajalingam, Sokkalingam; Daud, Hanita; Daud, Muhamad Azlan
2016-11-01
This paper presents a four point block one-step method for solving directly boundary value problems (BVP) with Neumann boundary conditions and Singular Perturnbation BVPs. This method is formulated using Lagrange interpolating polynomial. The block method will solve the second order linear Neumann and Singular Perturbation BVPs directly without reducing it to the system of first order differential equations. This direct method will compute four points simultaneously within a block using constant step size. This method will be used together with linear shooting technique to construct the solutions. The implementation is based on the predictor and corrector formulas. Numerical results are given to show the efficiency of this method compared to the existing methods.
Solution matching for a three-point boundary-value problem on atime scale
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Martin Eggensperger
2004-07-01
Full Text Available Let $mathbb{T}$ be a time scale such that $t_1, t_2, t_3 in mathbb{T}$. We show the existence of a unique solution for the three-point boundary value problem $$displaylines{ y^{DeltaDeltaDelta}(t = f(t, y(t, y^Delta(t, y^{DeltaDelta}(t, quad t in [t_1, t_3] cap mathbb{T},cr y(t_1 = y_1, quad y(t_2 = y_2, quad y(t_3 = y_3,. }$$ We do this by matching a solution to the first equation satisfying a two-point boundary conditions on $[t_1, t_2] cap mathbb{T}$ with a solution satisfying a two-point boundary conditions on $[t_2, t_3] cap mathbb{T}$.
C*-Algebra approach to the index theory of boundary value problems
Melo, Severino; Schick, Thomas
2012-01-01
Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported $K$-theory classes on the cotangent bundle over the interior of the manifold. The Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the symbols. In this paper we survey previous work how C*-algebra K-theory can be used to give...
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Thanin Sitthiwirattham
2012-01-01
Full Text Available By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Δ2u(t-1+a(tf(u(t=0, t∈{1,2,…,T}, u(0=β∑s=1ηu(s, u(T+1=α∑s=1ηu(s, where f is continuous, T≥3 is a fixed positive integer, η∈{1,2,...,T-1}, 0<α<(2T+2/η(η+1, 0<β<(2T+2-αη(η+1/η(2T-η+1, and Δu(t-1=u(t-u(t-1. We show the existence of at least one positive solution if f is either superlinear or sublinear.
Numerical continuation methods for dynamical systems path following and boundary value problems
Krauskopf, Bernd; Galan-Vioque, Jorge
2007-01-01
Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel''s 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects ...
THE HILBERT BOUNDARY VALUE PROBLEM FOR GENERALIZED ANALYTIC FUNCTIONS IN CLIFFORD ANALYSIS
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Zhongwei SI; Jinyuan DU
2013-01-01
Let R0,n be the real Clifford algebra generated by e1,e2,…,en satisfying eiej +ejei =-2δij,i,j =1,2,…,n.e0 is the unit element.Let Ω be an open set.A function f is called left generalized analytic in Ω if f satisfies the equation Lf =0,(0.1)where L =q0e0(δ)x0 + q1e1(δ)x1 + … +qnen(δ)xn,qi ＞ 0,i =0,1,…,n.In this article,we first give the kernel function.for the generalized analytic function.Further,the Hilbert boundary value problem for generalized analytic functions in Rn+1+ will be investigated.
A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems
Pan, Kejia; He, Dongdong; Hu, Hongling; Ren, Zhengyong
2017-09-01
In this paper, we develop a new extrapolation cascadic multigrid method, which makes it possible to solve three dimensional elliptic boundary value problems with over 100 million unknowns on a desktop computer in half a minute. First, by combining Richardson extrapolation and quadratic finite element (FE) interpolation for the numerical solutions on two-level of grids (current and previous grids), we provide a quite good initial guess for the iterative solution on the next finer grid, which is a third-order approximation to the FE solution. And the resulting large linear system from the FE discretization is then solved by the Jacobi-preconditioned conjugate gradient (JCG) method with the obtained initial guess. Additionally, instead of performing a fixed number of iterations as used in existing cascadic multigrid methods, a relative residual tolerance is introduced in the JCG solver, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple method based on the midpoint extrapolation formula is proposed to achieve higher-order accuracy on the finest grid cheaply and directly. Test results from four examples including two smooth problems with both constant and variable coefficients, an H3-regular problem as well as an anisotropic problem are reported to show that the proposed method has much better efficiency compared to the classical V-cycle and W-cycle multigrid methods. Finally, we present the reason why our method is highly efficient for solving these elliptic problems.
Hybrid state vector methods for structural dynamic and aeroelastic boundary value problems
Lehman, L. L.
1982-01-01
A computational technique is developed that is suitable for performing preliminary design aeroelastic and structural dynamic analyses of large aspect ratio lifting surfaces. The method proves to be quite general and can be adapted to solving various two point boundary value problems. The solution method, which is applicable to both fixed and rotating wing configurations, is based upon a formulation of the structural equilibrium equations in terms of a hybrid state vector containing generalized force and displacement variables. A mixed variational formulation is presented that conveniently yields a useful form for these state vector differential equations. Solutions to these equations are obtained by employing an integrating matrix method. The application of an integrating matrix provides a discretization of the differential equations that only requires solutions of standard linear matrix systems. It is demonstrated that matrix partitioning can be used to reduce the order of the required solutions. Results are presented for several example problems in structural dynamics and aeroelasticity to verify the technique and to demonstrate its use. These problems examine various types of loading and boundary conditions and include aeroelastic analyses of lifting surfaces constructed from anisotropic composite materials.
Valent, Tullio
1988-01-01
In this book I present, in a systematic form, some local theorems on existence, uniqueness, and analytic dependence on the load, which I have recently obtained for some types of boundary value problems of finite elasticity. Actually, these results concern an n-dimensional (n ~ 1) formal generalization of three-dimensional elasticity. Such a generalization, be sides being quite spontaneous, allows us to consider a great many inter esting mathematical situations, and sometimes allows us to clarify certain aspects of the three-dimensional case. Part of the matter presented is unpublished; other arguments have been only partially published and in lesser generality. Note that I concentrate on simultaneous local existence and uniqueness; thus, I do not deal with the more general theory of exis tence. Moreover, I restrict my discussion to compressible elastic bodies and I do not treat unilateral problems. The clever use of the inverse function theorem in finite elasticity made by STOPPELLI [1954, 1957a, 1957b]...
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A.S. BERDYSHEV; A. CABADA; B.Kh. TURMETOV
2014-01-01
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
On Solutions of the Integrable Boundary Value Problem for KdV Equation on the Semi-Axis
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Ignatyev, M. Yu., E-mail: ignatievmu@info.sgu.ru [Saratov State University, Department of Mathematics (Russian Federation)
2013-03-15
This paper is concerned with the Korteweg-de Vries (KdV) equation on the semi-axis. The boundary value problem with inhomogeneous integrable boundary conditions is studied. We establish some characteristic properties of solutions of the problem. Also we construct a wide class of solutions of the problem using the inverse spectral method.
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Gregory A. CHECHKIN; Rustem R. GADYL'SHIN
2007-01-01
We consider boundary-value problems with rapidly alternating types of boundary condi- tions. We present the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.
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Dongyuan Liu
2015-01-01
Full Text Available We consider the following state dependent boundary-value problem D0+αy(t-pD0+βg(t,y(σ(t+f(t,y(τ(t=0, 0
Sprlak, M.; Novak, P.; Pitonak, M.; Hamackova, E.
2015-12-01
Values of scalar, vectorial and second-order tensorial parameters of the Earth's gravitational field have been collected by various sensors in geodesy and geophysics. Such observables have been widely exploited in different parametrization methods for the gravitational field modelling. Moreover, theoretical aspects of these quantities have extensively been studied and are well understood. On the other hand, new sensors for observing gravitational curvatures, i.e., components of the third-order gravitational tensor, are currently under development. This fact may be documented by the terrestrial experiments Dulkyn and Magia, as well as by the proposal of the gravity-dedicated satellite mission called OPTIMA. As the gravitational curvatures represent new types of observables, their exploitation for modelling of the Earth's gravitational field is a subject of this study. Firstly, we derive integral transforms between the gravitational potential and gravitational curvatures, i.e., we find analytical solutions of the boundary value problems with gravitational curvatures as boundary conditions. Secondly, properties of the corresponding Green kernel functions are studied in the spatial and spectral domains. Thirdly, the correctness of the new analytical solutions is tested in a simulation study. The presented mathematical apparatus reveal important properties of the gravitational curvatures. It also extends the Meissl scheme, i.e., an important theoretical paradigm that relates various parameters of the Earth's gravitational field.
Interior and exterior solutions for boundary value problems in composite elastic and viscous media
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D. L. Jain
1985-01-01
Full Text Available We present the solutions for the boundary value problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties. The solutions are obtained inside both the guest and host media by an integral equation technique. The boundaries considered are an oblong, a triaxial ellipsoid and an elliptic cyclinder of a finite height and their limiting configurations in two and three dimensions. The exact interior and exterior solutions for an ellipsoidal inclusion and its limiting configurations are presented when the infinite host medium is subjected to a uniform strain. In the case of an oblong or an elliptic cylinder of finite height the solutions are approximate. Next, we present the formula for the energy stored in the infinite host medium due to the presence of an arbitrary symmetrical void in it. This formula is evaluated for the special case of a spherical void. Finally, we analyse the change of shape of a viscous incompressible ellipsoidal region embedded in a slowly deforming fluid of a different viscosity. Two interesting limiting cases are discussed in detail.
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Min Jia
2012-01-01
Full Text Available We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system -tαx(t=f(t,x(t,x'(t,x”(t,…,x(n-2(t, 0
Acceleration of multiple solution of a boundary value problem involving a linear algebraic system
Gazizov, Talgat R.; Kuksenko, Sergey P.; Surovtsev, Roman S.
2016-06-01
Multiple solution of a boundary value problem that involves a linear algebraic system is considered. New approach to acceleration of the solution is proposed. The approach uses the structure of the linear system matrix. Particularly, location of entries in the right columns and low rows of the matrix, which undergo variation due to the computing in the range of parameters, is used to apply block LU decomposition. Application of the approach is considered on the example of multiple computing of the capacitance matrix by method of moments used in numerical electromagnetics. Expressions for analytic estimation of the acceleration are presented. Results of the numerical experiments for solution of 100 linear systems with matrix orders of 1000, 2000, 3000 and different relations of variated and constant entries of the matrix show that block LU decomposition can be effective for multiple solution of linear systems. The speed up compared to pointwise LU factorization increases (up to 15) for larger number and order of considered systems with lower number of variated entries.
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Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations
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H. Batarfi
2015-01-01
Full Text Available We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.
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Omer Kelesoglu
2014-01-01
Full Text Available Adomian decomposition method (ADM is applied to linear nonhomogeneous boundary value problem arising from the beam-column theory. The obtained results are expressed in tables and graphs. We obtain rapidly converging results to exact solution by using the ADM. This situation indicates that the method is appropriate and reliable for such problems.
Amirkhanov, I V; Sarker, N R; Sarhadov, I
2004-01-01
In this work the solutions of different boundary-value problems are retrieved analytically and numerically for the equation of high order with small parameter at a higher derivative. The analysis of these solutions is given. It is found that for some variants of symmetric boundary conditions the solutions of a boundary-value problem for the equations of the 4th, 6th, $\\ldots$ orders transfer into the solution of a Schrödinger equation at $\\varepsilon \\to 0$ ($\\varepsilon $ is small parameter). The retrieved solutions with different knots are orthogonal among themselves. The results of numerical calculations are given.
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D. Goos
2015-01-01
Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
Beretta, Elena; Manzoni, Andrea; Ratti, Luca
2017-03-01
In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection of small inhomogeneities located inside a domain Ω , where the coefficients of the equation are altered, starting from observations of the solution of the equation on the boundary \\partial Ω . Exploiting theoretical results recently achieved in [13], we implement a reconstruction procedure based on the computation of the topological gradient of a suitable cost functional. Numerical results obtained for several test cases finally assess the feasibility and the accuracy of the proposed technique.
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侯祥林; 翟中海; 郑莉; 刘铁林
2012-01-01
针对非线性偏微分方程初边值问题,基于差分法和动态设计变量优化算法原理,以时间计算层上离散节点的未知函数值为设计变量,以离散节点的差分方程组构造程式化的目标函数,提出了离散节点处未知函数值的逐层高精度优化算法.编制通用程序求解具体典型算例.并通过与解析解对比,表明了求解方法的正确性和有效性,为广泛的工程应用提供条件.%For non-linear partial differential equations with initial-boundary value problems,based on the difference method and the optimization method with dynamic design variables,using unknown function values on discrete node points on time layer as design variables,the difference equations sets of all the discrete node points are constructed as stylized objective function.A layered accurate optimization algorithm about computing unknown function value on discrete node point is proposed.Universal computing program is designed,and practical examples are analyzed.Through comparing computation results with exact results,the effectiveness and the feasibility of proposed method are verified.The method can provide the condition for engineering application.
A Kernel-free Boundary Integral Method for Elliptic Boundary Value Problems ⋆
Ying, Wenjun; Henriquez, Craig S.
2013-01-01
This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green's functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GM-RES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong. PMID:23519600
On a new nonlocal boundary value problem for an equation of the mixed parabolic-hyperbolic type
Dildabek, Gulnar
2016-12-01
In this work a new nonlocal boundary value problem for an equation of the mixed type is formulated. This equation is parabolic-hyperbolic and belongs to the first kind because the line of type change is not a characteristic of the equation. Non-local condition binds points on boundaries of the parabolic and hyperbolic parts of the domain with each other. This problem is generalization of the well-known problems of Frankl type. A boundary value problem for the heat equation with conditions of the Samarskii-Ionlin type arises in solving this problem. Unlike the existing publications of the other authors related to the theme it is necessary to note that in this papers the nonlocal problems were considered in rectangular domains. But in our formulation of the problem the hyperbolic part of the domain coincides with a characteristic triangle. Unique strong solvability of the formulated problem is proved.
A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem
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Musa Çakır
2010-01-01
Full Text Available The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameter ε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.
POSlTlVE SOLUTl0NS 0F A FOURTH 0RDER BOUNDARY VALUE PROBLEM
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RenLishun
2003-01-01
The existence of positive solutions of the nonlinear fourth order problem u(4)(x)=λα（x)f(u(x)),u(0)=u′（0)=u′（1)=um(1)=0 is studied,where a:[0,1]→R may change sign,f(0)>0,λ>0 is sufficiently small.Our approach is based on the Leray-Schauder fixed point theorem.
On the Beam Functions Spectral Expansions for Fourth-Order Boundary Value Problems
Papanicolaou, N. C.; Christov, C. I.
2007-10-01
In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonlinear problems. We make use of the formulas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10-12 for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10-9. The beam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former.
The Third Initial-boundary Value Problem for a Class of Parabolic Monge-Ampère Equations
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Lü BO-QIANG; LI FENG-QUAN
2012-01-01
For the more general parabolic Monge-Ampère equations defined by the operator F(D2u + σ(x)),the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the equation are established.A new structure condition which is used to get a priori estimate is established.
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Baoqiang Yan
2006-07-01
Full Text Available In this paper, Krasnoselskii's theorem and the fixed point theorem of cone expansion and compression are improved. Using the results obtained, we establish the existence of multiple positive solutions for the singular second-order boundary-value problems with derivative dependance on finite and infinite intervals.
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无
2006-01-01
Dirichlet boundary value problems for perturbed second-order differential equations on a half line are investigated in this paper. The methods mainly depend on the calculus of variations to the classical functionals. Sufficient conditions are obtained for the existence of the solutions.
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Qing-liu Yao
2003-01-01
The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point theorem on the cone.
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GUO Fei
2007-01-01
In this paper we study the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems in the domain D={(t,x)| t≥O,x≥O}.Under the assumption that the source term satisfies the matching condition,a sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.
Dambrine, M.; Greff, I.; Harbrecht, H.; Puig, B.
2017-02-01
The present article is dedicated to the numerical solution of homogeneous Neumann boundary value problems on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on the random domain can be transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Ventcell boundary condition and yields a second order accurate solution in the scale parameter ε of the layer's thickness. With the help of the Karhunen-Loève expansion, we transform this random boundary value problem into a deterministic, parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter ε. Numerical results validate our theoretical findings.
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无
2009-01-01
In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.
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Xing-qiu Zhang; Jing-xian Sun
2009-01-01
By constructing a special cone and using cone compression and expansion fixed point theorem,this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for a class of first order differential equation.As applications of the main results,two examples are given at the end of this paper.
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王同科
2002-01-01
In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs fromthe high order generalized difference methods. It is proved that the method has optimal order er-ror estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.
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Vrabel Robert
2011-01-01
Full Text Available Abstract This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.
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Zhenlai Han
2012-11-01
Full Text Available In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t+a(tf(u=0, quad 0
On Solution of the Integrable Initial Boundary Value Problem for KdV Equation on the Semi-axis
Energy Technology Data Exchange (ETDEWEB)
Ignatyev, Mikhail Yurievich, E-mail: mikkieram@gmail.com [Saratov State University, Department of Mathematics (Russian Federation)
2013-12-15
This paper is concerned with the Korteweg-de Vries (KdV) equation on the right semi-axis. The initial boundary value problem with inhomogeneous integrable boundary conditions is studied. We show that, under some conditions on the initial data the problem has a solution and provide the procedure for constructing this solution. The procedure is based on the inverse spectral method and consists of several steps reducing to either solving some linear problems or calculations via some explicit formulas.
Lomtatidze, A. (Alexander)
2016-01-01
The aim of the present article is to get efficient conditions for the solvability of the periodic boundary value problém $$ u''=f(t,u);quad u(0)=u(omega),;; u'(0)=u'(omega), $$ where the function $fcolon[0,omega]times,]0,+infty[,tobbr$ satisfies local Ca-ra-th'{e}o-do-ry conditions, i.e., it may have ''singularity'' for $u=0$. For this purpose, first the technique of differential inequalities is developed and the question on existence and uniqueness of a~positive solution of the linear problé...
Boundary value problem for one-dimensional fractional differential advection-dispersion equation
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Khasambiev Mokhammad Vakhaevich
2014-07-01
Full Text Available An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.
Institute of Scientific and Technical Information of China (English)
TAN Hui-xuan; FENG Han-ying; FENG Xing-fang; DU Ya-tao
2015-01-01
In this paper, we consider the three-point boundary value problem (φp(uˊˊ(t)))ˊ +a(t)f(t, u(t), uˊ(t), uˊˊ(t)) = 0, t ∈ [0, 1] subject to the boundary conditions u(0) =βuˊ(0), uˊ(1) = αuˊ(η), uˊˊ(0) = 0, where φp(s) = |s|p−2s with p > 1, 0 < α, η < 1 and 0 ≤ β < 1. Applying a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem.
Baur, Benedict
2014-01-01
Benedict Baur presents modern functional analytic methods for construction and analysis of Feller processes in general and diffusion processes in particular. Topics covered are: Construction of Lp-strong Feller processes using Dirichlet form methods, regularity for solutions of elliptic boundary value problems, construction of elliptic diffusions with singular drift and reflection, Skorokhod decomposition and applications to Mathematical Physics like finite particle systems with singular interaction. Emphasize is placed on the handling of singular drift coefficients, as well as on the discussion of pointwise and pathwise properties of the constructed processes rather than just the quasi-everywhere properties commonly known from the general Dirichlet form theory. Contents Construction of Lp-Strong Feller Processes Elliptic Boundary Value Problems Skorokhod Decomposition for Reflected Diffusions with Singular Drift Particle Systems with singular interaction Target Groups Graduate and PhD students, researchers o...
Institute of Scientific and Technical Information of China (English)
ZHANG Yu-chuan; ZHOU Zong-fu
2014-01-01
In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian(Φp(u000))0(t)+a(t)f(t, u(t−τ), u0(t))=0, 0
Finster, Felix
2015-01-01
We consider a boundary value problem for the Dirac equation in a four-dimensional, smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the space-time includes horizons, where the Hamiltonian fails to be elliptic.
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Fenghua Miao
2013-07-01
Full Text Available In this article, we study the fractional q-difference boundary-value problems with p-Laplacian operator $$displaylines{ D_{q}^{gamma}(phi_p(D_{q}^{alpha}u(t + f(t,u(t=0, quad 0 1. By using a fixed-point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions.
Schmidt, E. J. P. Georg
This paper is concerned with the total—and strict total—positivity of the integral kernels which relate the solutions of parabolic initial boundary value problems to the initial and boundary data. Maximum principle arguments, in conjunction with a characterisation of total positivity in terms of variation diminishing properties, are used to prove total positivity of both the initial value and boundary value kernels under mild assumptions; similar methods yield strict total positivity when the coefficients appearing in the equation are analytic in either the time or the space variable. In this way we extend results of Karlin and McGregor on the total positivity of the kernel associated with the initial data; their proofs used the determinantal definition of total positivity and exploited the work of Gohberg and Krein on the Green's functions of Sturm-Liouville operators.
Positive Solutions of Sub-Linear Semi-Positone Boundary Value Problem System%次线性半正微分边值系统的正解
Institute of Scientific and Technical Information of China (English)
徐西安
2008-01-01
In this paper,we study the existence of positive solutions of a sub-linear semi-positone differential boundary value problems system with positive parameter.We prove that the semipositone differential boundary value problems system has at least one positive solution for the parameter sufficiently large.
Institute of Scientific and Technical Information of China (English)
闫宝强
2011-01-01
This paper discusses singular three-point boundary value problems y″(t) + a(t)f (t, y(t), y′(t)) = 0, 0 ＜ t＜ 1, y′(0) = 0, y(1) = αy(η),where 0 ＜ α ＜ 1, 0 <η < 1, f changes sign and may be singular at y = 0 and y′ = 0%该文讨论奇异三点边值问题y"(t)+a(t)f(t,y(t),y'(t))=0,0＜t＜1,y'(0)=0,y(1)=αy(η)正解的存在与不存在性,这里0＜α＜1,0＜η＜1,f变号且在y=0和y'=0具有奇异性.
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Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
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Johnny Henderson
2016-01-01
Full Text Available We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.
Application of Two-Parameter Extrapolation for Solution of Boundary-Value Problem on Semi-Axis
Zhidkov, E P
2000-01-01
A method for refining approximate eigenvalues and eigenfunctions for a boundary-value problem on a half-axis is suggested. To solve the problem numerically, one has to solve a problem on a finite segment [0,R] instead of the original problem on the interval [0,\\infty). This replacement leads to eigenvalues' and eigenfunctions' errors. To choose R beforehand for obtaining their required accuracy is often impossible. Thus, one has to resolve the problem on [0,R] with larger R. If there are two eigenvalues or two eigenfunctions that correspond to different segments, the suggested method allows one to improve the accuracy of the eigenvalue and the eigenfunction for the original problem by means of extrapolation along the segment. This approach is similar to Richardson's method. Moreover, a two-parameter extrapolation is described. It is combination of the extrapolation along the segment and Richardson's extrapolation along a discretization step.
Extended cubic B-spline method for solving a linear system of second-order boundary value problems.
Heilat, Ahmed Salem; Hamid, Nur Nadiah Abd; Ismail, Ahmad Izani Md
2016-01-01
A method based on extended cubic B-spline is proposed to solve a linear system of second-order boundary value problems. In this method, two free parameters, [Formula: see text] and [Formula: see text], play an important role in producing accurate results. Optimization of these parameters are carried out and the truncation error is calculated. This method is tested on three examples. The examples suggest that this method produces comparable or more accurate results than cubic B-spline and some other methods.
Institute of Scientific and Technical Information of China (English)
范天佑; 郭玉翠
1997-01-01
The mathematical theory of elasticity for planar pentagonal quasicrystals is developed and some analytic solutions for a class of mixed boundary-value problems (corresponding to a Griffith crack) of the theory are offered.An alternate procedure and a direct integral approach are proposed.Some analytical solutions are constructed and the stress and displacement fields of a Griffith crack in the quasicrystals are determined.A basis for further studying the mechanical behavior of the material related to planar defects is provided.
On the solvability of a boundary value problem on the real line
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Marcelli Cristina
2011-01-01
Full Text Available Abstract We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations ( a ( x Φ ( x ′ ( t ′ = f ( t , x ( t , x ′ ( t , a . e . t ∈ ℝ governed by a nonlinear differential operator Φ extending the classical p-Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f ( t , ⋅ , ⋅ ≈ 1 t . We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x' = b(t, xc(x, x'. Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.
On a difference scheme for nonlocal heat transfer boundary-value problem
Akhymbek, Meiram E.; Sadybekov, Makhmud A.
2016-08-01
In this paper, we propose a new method of solving nonlocal problems for the heat equation with finite difference method. The main important feature of these problems is their non-self-adjointness. This non-self-adjointness causes major difficulties in their analytical and numerical solving. The problems, which boundary conditions do not possess strong regularity, are less studied. The scope of study of the paper justifies possibility of building a stable difference scheme with weights for abovementioned type of problems.
Koch曲线上的齐次Riemann边值问题%Homogeneous Riemann Boundary Value Problems for Koch Curve
Institute of Scientific and Technical Information of China (English)
阮正顺; 罗艾花
2012-01-01
当L为典型的分形曲线-Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.%In this paper, when Lis substituted for Koch curve, Riemann boundary value problems was defined, but generally speaking, Cauchy-type integral is meaningless on Koch curve. When some analytic conditions are attached to functions G(z)and g(z), through the limit function of a sequence of Cauchy-type integrals, the homogeneous Riemann boundary problems on Koch curve are introduced, some similar results was attained with the classical boundary value problems for analytic functions.
Bifurcation of solutions of nonlinear Sturm–Liouville problems
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Gulgowski Jacek
2001-01-01
Full Text Available A global bifurcation theorem for the following nonlinear Sturm–Liouville problem is given Moreover we give various versions of existence theorems for boundary value problems The main idea of these proofs is studying properties of an unbounded connected subset of the set of all nontrivial solutions of the nonlinear spectral problem , associated with the boundary value problem , in such a way that .
Holota, Petr; Nesvadba, Otakar
2014-05-01
In geodesy mathematical techniques for gravity field studies that rest on the concept of the so-called classical solution of boundary value problems, have a rather traditional position. Nevertheless, the range of the tools for treating problems in this field is much wider. For instance the concept of the weak solution met with a considerable attention. From this point of view the approach is associated with constructing the respective integral kernels or Green's function in case we consider the classical solution concept or with the choice and constructing basis functions in case we are lucking for the weak solution of the problem. Within the tools considered we discuss also the use of reproducing kernels. In both the cases (classical or weak) the construction of the apparatus above represents and important technical step. It is not elementary, but for a number of fundamental boundary value problems the solution is known, in particular in the case of a spherical solution domain. The sphere, however, is rather far from the real shape of the Earth, which is interpreted here in terms of a functional analytic norm. The distance has a negative effect on any attempt to reach the solution of the boundary value problems considered (and to bridge the departure of the Earth's surface from the sphere) by an iteration procedure based on a successive application of a solution technique developed for the spherical boundary. From this point of view the construction of the integral kernels and basis functions for an oblate ellipsoid of revolution means a step closer towards reality. In this contribution we on the one hand give an overview of the results already achieved and subsequently develop the topic. The summation of series of ellipsoidal harmonics is one of the key problems in this connection. Hypergeometric functions and series are applied too. We also show where the use of Legendre elliptic integrals adds to the solution of the problem. It is interesting that they do not
Multiple Positive Solutions to Third-Order Three-Point Singular Semipositone Boundary Value Problem
Indian Academy of Sciences (India)
Huimin Yu; L Haiyan; Yansheng Liu
2004-11-01
By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple positive solutions for the third-order three-point singular semipositone BVP: \\begin{equation*}\\begin{cases}x'"(t)- f(t,x)=0, & t\\in(0, 1);\\\\ x(0)=x'()=x"(1)=0,\\end{cases}\\end{equation*} where $\\frac{1}{2} < < 1$, the non-linear term $f(t,x): (0,1)×(0,=∞)→(-∞ +∞)$ is continuous and may be singular at = 0, = 1, and = 0, also may be negative for some values of and , is a positive parameter.
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Xiaoyong Xu
2015-01-01
Full Text Available A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs and initial value problems (IVPs in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures.
Chen, Wan; Wetton, Brian
2009-02-01
We consider a free boundary-value problem based on a simplified model of two-phase flow in porous media. The model has two independent variables on each side of the free interface. At the interface at steady state, five mixed Dirichlet and Neumann conditions are given. The movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady-state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). Following previous work, we consider the use of other residuals for the normal velocity that have superior numerical properties. The well-posedness criteria for this vector example are particularly clear. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in numerical computations. Although the finite-difference implementation in this work is not applicable to general problems, it has superior performance to previous implementations.
Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity
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Sergey I. Senashov
2012-10-01
Full Text Available For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for quasilinear system and the problem for conservation laws system permits to construct the characteristic lines in domains, where Jacobian of hodograph transformations is equal to zero. Moreover, the conservation laws give all solutions of the linearized system. Some examples from the gas dynamics and theory of plasticity are considered.
Optimization of the Forcing Term for the Solution of Two-Point Boundary Value Problems
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Gianni Arioli
2013-01-01
by standard methods of constrained optimization, for example, with Lagrange multipliers. We provide an application of this algorithm to the planar restricted three body problem in order to study the planning of low-thrust transfer orbits.
LIMIT BEHAVIOUR OF SOLUTIONS TO EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
LiFengquan
2002-01-01
In this paper,we discuss the limit behaviour of solutions to equivalued surface boundayr value problem for parabolic equatiopns when the equivalued surface boundary shriks to a point and the space dimension of the domain is two or more.
Monotone Iterative Method for Set-valued Quasilinear Elliptic Boundary Value Problems
Institute of Scientific and Technical Information of China (English)
SUN Le-lin; XU Di-hong; CHENG Jian; Li Rong-hua
2001-01-01
@@1. Problem and Assumptions This paper deals with the solutions of the following differential inclusion problem: Au ∈f(x,u), x ∈ Ω; (1) u =0, x∈ Ω, whereAu(x)=Ω RN is a bounded domain with piecewise Lipschitz boundary Ω , Du = (D1 u, D2 u,…,DNu), Diu = , i = 1,2,…, N, and f: Ω × R→2R is a set-valued unction.
An accurate boundary value problem solver applied to scattering from cylinders with corners
Helsing, Johan
2012-01-01
In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr\\"om scheme where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we specialize to scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and d the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder.
Boundary value problems for the nd-order Seiberg-Witten equations
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Doria Celso Melchiades
2005-01-01
Full Text Available It is shown that the nonhomogeneous Dirichlet and Neuman problems for the nd-order Seiberg-Witten equation on a compact -manifold admit a regular solution once the nonhomogeneous Palais-Smale condition is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace of configuration space. The coercivity of the -functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of -norms of spinor solutions and the gauge fixing lemma.
Boundary Value Technique for Initial Value Problems Based on Adams-Type Second Derivative Methods
Jator, S. N.; Sahi, R. K.
2010-01-01
In this article, we propose a family of second derivative Adams-type methods (SDAMs) of order up to 2k + 2 ("k" is the step number) for initial value problems. The methods are constructed through a continuous approximation of the SDAM which is obtained by multistep collocation. The continuous approximation is used to obtain initial value methods,…
Towards parameter limits of displacement boundary value problems for Mohr-Coulomb models
Rohe, A.
2013-01-01
To solve problems in geotechnical engineering often numerical methods such as the Finite Element Method (FEM) are used. This method can be applied for example for the calculation of the strength of dikes, the determination of the stability of (rail)road embankments, the prediction of deformations du
The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates
DEFF Research Database (Denmark)
Grubb, Gerd
2011-01-01
For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ+ of the boundary and a Dirichlet condition on the other part Σ−. We show a Kreĭn resolvent formula, where the difference between its r...
Initial-boundary value problems for second order systems of partial differential equations
Kreiss, Heinz-Otto; Petersson, N Anders
2010-01-01
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order system. Here we will use the theory of pseudo-differential operators combined with mode analysis. There are many desirable properties of this approach: 1) The reduction to first order systems of pseudo-differential equations poses no difficulty and always gives a system of 2n equations. 2) We can localize the problem, i.e., it is only necessary to study the Cauchy problem and halfplane problems with constant coefficients. 3) The class of problems we can treat is much larger than previous approaches based on "integ...
ASYMPTOTICS OF INITIAL BOUNDARY VALUE PROBLEMS OF BIPOLAR HYDRODYNAMIC MODEL FOR SEMICONDUCTORS
Institute of Scientific and Technical Information of China (English)
Ju Qiangchang
2004-01-01
In this paper, we study the asymptotic behavior of the solutions to the bipolar hydrodynamic model with Dirichlet boundary conditions. It is shown that the initial boundary problem of the model admits a global smooth solution which decays to the steady state exponentially fast.
Fayolle, Guy; Malyshev, Vadim
2017-01-01
This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function Theory, Boundary Value Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems. Part II borrows spec...
Jiu, Quansen; Liu, Jitao; Wu, Jiahong; Yu, Huan
2017-10-01
This paper focuses on the initial- and boundary-value problem for the two-dimensional micropolar equations with only angular velocity dissipation in a smooth bounded domain. The aim here is to establish the global existence and uniqueness of solutions by imposing natural boundary conditions and minimal regularity assumptions on the initial data. Besides, the global solution is shown to possess higher regularity when the initial datum is more regular. To obtain these results, we overcome two main difficulties: one due to the lack of full dissipation and one due to the boundary conditions. In addition to the global regularity problem, we also examine the large time behavior of solutions and obtain explicit decay rates.
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Jiang Zhengxian
2009-01-01
Full Text Available This paper deals with a two-point boundary value problem of a class of Duffing-type systems with non-C1 perturbation term. Several existence and uniqueness theorems were presented.
Convergence of a continuous BGK model for initial boundary-value problems for conservation laws
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Driss Seghir
2001-11-01
Full Text Available We consider a scalar conservation law in the quarter plane. This equation is approximated in a continuous kinetic Bhatnagar-Gross-Krook (BGK model. The convergence of the model towards the unique entropy solution is established in the space of functions of bounded variation, using kinetic entropy inequalities, without special restriction on the flux nor on the equilibrium problem's data. As an application, we establish the hydrodynamic limit for a $2imes2$ relaxation system with general data. Also we construct a new family of convergent continuous BGK models with simple maxwellians different from the $chi$ models.
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Т. Horsin
2014-01-01
Full Text Available We consider an optimal control problem associated to Dirichlet boundary valueproblem for linear elliptic equations on a bounded domain Ω. We take the matrixvalued coecients A(x of such system as a control in L1(Ω;RN RN. One of the important features of the admissible controls is the fact that the coecient matrices A(x are non-symmetric, unbounded on Ω, and eigenvalues of the symmetric part Asym = (A + At=2 may vanish in Ω.
Tatsii, R. M.; Pazen, O. Yu.
2016-03-01
A constructive scheme for the construction of a solution of a mixed problem for the heat conduction equation with piecewise-continuous coefficients coordinate-dependent in the final interval is suggested and validated in the present work. The boundary conditions are assumed to be most general. The scheme is based on: the reduction method, the concept of quasi-derivatives, the currently accepted theory of the systems of linear differential equations, the Fourier method, and the modified method of eigenfunctions. The method based on this scheme should be related to direct exact methods of solving mixed problems that do not employ the procedures of constructing Green's functions or integral transformations. Here the theorem of eigenfunction expansion is adapted for the case of coefficients that have discontinuity points of the 1st kind. The results obtained can be used, for example, in investigating the process of heat transfer in a multilayer slab under conditions of ideal thermal contact between the layers. A particular case of piecewise-continuous coefficients is considered. A numerical example of calculation of a temperature field in a real four-layer building slab under boundary conditions of the 3rd kind (conditions of convective heat transfer) that model the phenomenon of fire near one of the external surfaces is given.
Boundary value problems for the 2nd-order Seiberg-Witten equations
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Celso Melchiades Doria
2005-02-01
Full Text Available It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition Ã¢Â„Â‹ is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace Ã°ÂÂ’ÂžÃŽÂ±Ã¢Â„Â of configuration space. The coercivity of the Ã°ÂÂ’Â®Ã°ÂÂ’Â²ÃŽÂ±-functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of LÃ¢ÂˆÂž-norms of spinor solutions and the gauge fixing lemma.
Boundary Value Problem for Analysis of Portal Double-Row Stabilizing Piles
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Cheng Huang
2013-01-01
Full Text Available This paper presents a new numerical approach for computing the internal force and displacement of portal double-row piles used to stabilize potential landslide. First, the new differential equations governing the mechanical behaviour of the stabilizing pile are formulated and the boundary conditions are mathematically specified. Then, the problem is numerically solved by the high-accuracy Runge-Kutta finite difference method. A program package has been developed in MATLAB depending on the proposed algorithm. Illustrative examples are presented to demonstrate the validity of the developed program. In short, the proposed approach is a practical new idea for analyzing the portal double-row stabilizing pile as a useful supplement to traditional methods such as FEM.
Application of Splines to the Numerical Solution of Two-Point Boundary- Value Problems
1978-12-01
Computer Journal, Vol. 12, No. 2, May 1969, pp. 188-192. Kamke , E. Differentialgleichungen L~sungsmethoden and I.JSsungen, I, 6. Auflage, p. 558, Eq... KAMKE A R 0 0 8 6 7 1 | 0 : 2 9 WED MAY lOe 1978 ND I M JCASE N NT JOUT LPRNT LN(]IRN 2 5 0 | 51 3 0 0 | | 11 ] TOL RSC O, 0 5 t O 0 0 0 D - O...E D C - T R - 7 8 - 5 9 OC T 0 3 0 C O P S PROBLEM FROM KAMKE X I 0 , 0 2 1 . 3 3 o 0 0 - 0 1 3 2 , 0 6 0 8 3 - O l 4 3 . 0 6 3 3 D - 0
Institute of Scientific and Technical Information of China (English)
De-xiang Ma; Wei-gao Ge; Xue-gang Chen
2005-01-01
In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{(φp(u'))' + q(t)f(t,u,u') = 0, 0 ＜ t ＜ 1,{u(0)=n∑i=1αiu(ξi),u'(1)=n∑i=1βiu'(ξi),whereφp(s)=|s|p-2s, p≥2;ξi∈(0, 1)(i=1,2,...,n),0≤αi,βi＜1(i=1,2,...,n),0≤n∑i=1αi,n∑i=1β＜1,and q(t) may be singular at t= 0, 1; f(t, u, u') may be singular at u'= 0.
Hong, S. H.; Wilhelm, H. E.
1978-01-01
An electrical discharge between two ring electrodes embedded in the mantle of a cylindrical chamber is considered, in which the plasma in the anode and cathode regions rotates in opposite directions under the influence of an external axial magnetic field. The associated boundary-value problem for the coupled partial differential equations describing the azimuthal velocity and radial current-density fields is solved in closed form. The velocity, current density, induced magnetic induction, and electric fields are presented for typical Hartmann numbers, magnetic Reynolds numbers, and geometry parameters. The discharge is shown to produce anodic and cathodic plasma sections rotating at speeds of the order 1,000,000 cm/sec for conventional magnetic field intensities. Possible application of the magnetoactive discharge as a plasma centrifuge for isotope separation is discussed.
DEFF Research Database (Denmark)
Dankowicz, Harry; Schilder, Frank
This paper presents a collocation toolbox for multi-point, boundary-value problems. This toolbox has been recently developed by the authors to support general-purpose parameter continuation of sets of constrained orbit segments, such as i) segmented trajectories in hybrid dynamical systems......, for example, mechanical systems with impacts, friction, and switching control, ii) homoclinic orbits represented by an equilibrium point and a finite-time trajectory that starts and ends near this equilibrium point, and iii) collections of trajectories that represent quasi-periodic invariant tori...... the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations....
Institute of Scientific and Technical Information of China (English)
Xiaohong HAO; Zongfu ZHOU
2013-01-01
In this paper,we consider the following multi-point boundary value problem of fractional differential equation Dα0+u(t) =f(t,u(t),Dα-10+ u(t),Dα+20u(t),Dα-30u(t)),t ∈ (0,1),I4-α0+u(0) =0,Dα-10+u(0) =nΣi=1αi Dα-10+u(ξi),Dα-20+u(1) =nΣj=1 βjDα-20+ u(ηj),Dα-30+u(1)-Dα-30+u(0) =Dα-20+2u(1/2),where 3 ＜ α ≤ 4 is a real number.By applying Mawhin coincidence degree theory and constructing suitable operators,some existence results of solutions can be established.
Dujardin, G. M.
2009-08-12
This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas\\' transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over. © 2009 The Royal Society.
Holota, Petr; Nesvadba, Otakar
2017-04-01
The aim of this paper is to discuss the solution of the linearized gravimetric boundary value problem by means of the method of successive approximations. We start with the relation between the geometry of the solution domain and the structure of Laplace's operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. Laplace's operator has a relatively simple structure in terms of ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from an oblate ellipsoid of revolution, even if it is optimally fitted. Therefore, an alternative is discussed. A system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces is used. Clearly, the structure of Laplace's operator is more complicated in this case. It was deduced by means of tensor calculus and in a sense it represents the topography of the physical surface of the Earth. Nevertheless, the construction of the respective Green's function is more simple, if the solution domain is transformed. This enables the use of the classical Green's function method together with the method of successive approximations for the solution of the linear gravimetric boundary value problem expressed in terms of new coordinates. The structure of iteration steps is analyzed and where useful also modified by means of the integration by parts. Comparison with other methods is discussed.
Elliptic boundary value problems
Maz'ya, V G; Plamenevskii, B A; Stupyali, L; Plamenevskii, B A
1984-01-01
The papers in this volume have been selected, translated, and edited from publications not otherwise translated into English under the auspices of the AMS-ASL-IMS Committee on Translations from Russian and Other Foreign Languages.
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Taha Mohamed
2014-06-01
Full Text Available In the present work, the recursive differentiation method (RDM is introduced and implemented to obtain analytical solutions for differential equations governing different types of boundary value prob- lems (BVP. Then, the method is applied to investigate the static behaviour of a beam-column resting on a two parameter foundation subjected to different types of lateral loading. The analytical solutions obtained using RDM and Adomian decomposition method (ADM are found similar but the RDM requires less mathematical effort. It is indicated that the RDM is reliable, straightforward and efficient for investigation of BVP in finite domains. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. However, it is indicated that, in the case of beam-columns resting on foundations, the beam-column may be buckled in a higher mode rather than a lower one, then the critical load in that case is that accompanies the higher mode. This result is very important to avoid static instability as it is widely common that the buckling load of the first buckling mode is always the smaller one, which is true only in the case of the beam-columns without foundations.
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姚庆六
2008-01-01
By constructing suitable Banach space.an existence theorem is established under a condition of linear growth for the third-order boundary value problem u'"(t)+f(t,u(t),t'(t),u"(t))=0,0＜t＜1,u(0)=u'(0)=u'(1)=0,where the nonlinear term contains first and second derivatives of unknown function.In this theorem the nonlinear term f(t,u,v,w)may be singular at t=0 and t=1.The main ingredient is Leray-Schauder nonlinear alternative.
Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.
2012-10-01
A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
Institute of Scientific and Technical Information of China (English)
黄沙; 闻国椿
2000-01-01
This paper deals with the Riemann-Hilbert boundary value problem for general nonlinear elliptic complex equations of second order. Firstly we propose the Riemann-Hilbert problem and its well posedness, and then we give the representation of solutions for the modified boundary value problem and prove its solvability, and finally derive solvability conditions of the original Riemann-Hilbert problem.%讨论了一般二阶非线性椭圆复方程的Riemann-Hilbert边值问题. 首先给出Riemann-Hilbert问题及其适应性的概念,其次给出改进后的边值问题解的表述并证明了它的可解性,最后导出原Riemann-Hilbert边值问题的可解条件.
Hochman, Amit; White, Jacob K
2011-01-01
A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context of the 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystr\\"{o}m method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented.
Zhang, Xiaomin; Zhang, Long; Chu, Zhongxiang; Peng, Song
2016-09-01
In this paper, the periodic structure material is modeled as the continuum homogeneous micro-polar media subjecting to thermo-mechanical interaction. Meanwhile, a series of equivalent quantities such as the equivalent stress, couple stress, displacement gradient and torsion tensor were defined by the integral forms of the boundary values of the external surface force, moment, displacement and the angular displacement, and were proved to satisfy the equivalence conditions of virtual work. Based on above works, the displacement boundary value problem was established to deduce the equivalent constitutive equation. Assume the representative volume element is composed of the spatial cross-framework, and applying the boundary value problem of displacement on frame structures, the equivalent elastic coefficients, temperature coefficients of equivalent stress and the temperature gradient coefficients of equivalent couple stress are deduced. In addition, themethod can also be extended to the stress boundary value problem to deduce the equivalent constitutive equation. The calculations indicate that the equivalent result can be obtained from the two kinds of boundary value problems.
A Singular Quasilinear Elliptic Boundary Value Problem%关于一个拟线性椭圆边值问题的研究
Institute of Scientific and Technical Information of China (English)
王俊禹; 郑大伟; 高文杰
2003-01-01
A boundary value problem of a nonlinear partial differential equation with the N-dimensional p-Laplacian as the principal part is studied. The nonlinear term in the equation may have singularity and discontinuity in its variables. A typical form of the equation is the well-known Emden-Fowler equation with negative exponent. The existence and uniqueness results are obtained via a perturbation technique, the Schauder fixed point theorem and careful analysis of the property of the solutions.%本文研究一个主部为N维P-Laplace微分算子的非线性偏微分方程的边值问题.方程中的非线性项关于其变量可以具有奇性和间断性.这类方程的一种典型情形是大家熟知的具负指数的Emden-Fowler方程.本文利用摄动技巧,Schauder不动点定理,经对解的性质的精细分析,获得了解的存在性和唯一性结果.
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Vdovin, V.; Watari, T.; Fukuyama, A.
1996-12-01
We develop the theory for the wave excitation, propagation and absorption in 3-dimensional (3D) stellarator equilibrium high beta plasma in ion cyclotron frequency range (ICRF). This theory forms a basis for a 3D code creation, urgently needed for the ICRF heating scenarios development for the constructed LHD and projected W7-X stellarators and for the stellarators being at operation (like CHS, W7-AS, etc.). The theory solves the 3D Maxwell-Vlasov antenna-plasma-conducting shell boundary value problem in the non - orthogonal flux coordinates ({psi}, {theta}, {phi}), {psi} being magnetic flux function, {theta} and {phi} being the poloidal and toroidal angles, respectively. All basic physics, like wave refraction, reflection and diffraction are firstly self consistently included, along with the fundamental ion and ion minority cyclotron resonances, two ion hybrid resonance, electron Landau and TTMP absorption. Antenna reactive impedance and loading resistance are also calculated and urgently needed for an antenna -generator matching. This is accomplished in a real confining magnetic field being varying in a plasma major radius direction, in toroidal and poloidal directions, through making use of the hot dense plasma dielectric kinetic tensor. The theory is developed in a manner that includes tokamaks and magnetic mirrors as the particular cases through general metric tensor (provided by an equilibrium solver) treatment of the wave equations. We describe the structure of newly developed stellarator ICRF 3D full wave code STELION, based on theory described in this report. (J.P.N.)
Vagh, Hardik A.; Baghai-Wadji, Alireza
2008-12-01
Current technological challenges in materials science and high-tech device industry require the solution of boundary value problems (BVPs) involving regions of various scales, e.g. multiple thin layers, fibre-reinforced composites, and nano/micro pores. In most cases straightforward application of standard variational techniques to BVPs of practical relevance necessarily leads to unsatisfactorily ill-conditioned analytical and/or numerical results. To remedy the computational challenges associated with sub-sectional heterogeneities various sophisticated homogenization techniques need to be employed. Homogenization refers to the systematic process of smoothing out the sub-structural heterogeneities, leading to the determination of effective constitutive coefficients. Ordinarily, homogenization involves a sophisticated averaging and asymptotic order analysis to obtain solutions. In the majority of the cases only zero-order terms are constructed due to the complexity of the processes involved. In this paper we propose a constructive scheme for obtaining homogenized solutions involving higher order terms, and thus, guaranteeing higher accuracy and greater robustness of the numerical results. We present
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Nguyen Manh Hung
2008-03-01
Full Text Available In this paper, we consider the second initial boundary value problem for strongly general Schrodinger systems in both the finite and the infinite cylinders $Q_T, 0
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Bazalii, B V; Degtyarev, S P [Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (Ukraine)
2013-07-31
An elliptic boundary-value problem for second-order equations with nonnegative characteristic form is investigated in the situation when there is a weak degeneracy on the boundary of the domain. A priori estimates are obtained for solutions and the problem is proved to be solvable in some weighted Hölder spaces. Bibliography: 18 titles.
Institute of Scientific and Technical Information of China (English)
FANG Shao-mei; GUO Bo-ling
2005-01-01
The following initial-boundary value problem for the systems with multidimensional inhomogeneous generalized Benjamin-Bona-Mahony (GBBM)equations is reviewed. The existence of global attractors of this problem was proved by means of a uniform priori estimate for time.
一阶随机微分方程的边值问题%Boundary Value Problems for First Order Stochastic Differential Equations
Institute of Scientific and Technical Information of China (English)
王妍; 韩月才
2007-01-01
In this paper,we present a new technique to study nonlinear stochastic differential equations with periodic boundary value condition (in the sense of expectation).Our main idea is to decompose the stochastic process into a deterministic term and a new stochastic term with zero mean value.Then by using the contraction mapping principle and Leray-Schauder fixed point theorem,we obtain the existence theorem.Finally,we explain our main results by an elementary example.
Frederickson, P. O.; Wessel, W. R.
1979-01-01
Certain physical processes are modeled by partial differential equations which are parabolic over part of the domain and elliptic over the remainder. A family of semi-implicit algorithms which are well suited to initial-boundary value problems of this mixed type is discussed. One important feature of these algorithms is the use of an approximate inverse for the solution of the implicit linear system. A strong error analysis results in an estimate of the total error as a function of approximate inverse error e and time step h.
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Yanmei Sun
2012-01-01
Full Text Available By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.
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Abdelfatah Bouziani
2003-01-01
Full Text Available This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem.
Institute of Scientific and Technical Information of China (English)
袁益让; 李长峰
2012-01-01
可压缩可混溶油、水三维渗流动边值问题的研究,对重建盆地发育中油气资源运移、聚集的历史和评估油气资源的勘探与开发有重要的价值,其数学模型是一组非线性耦合偏微分方程组的动边值问题.该文对有界域的动边值问题提出一类新的二阶修正迎风差分格式,应用区域变换、变分形式、能量方法、差分算子乘积交换性理论、高阶差分算子的分解、微分方程先验估计的理论和技巧,得到了最佳l2误差估计结果.该方法已成功应用到油资评估的数值模拟中.它对这一领域的模型分析,数值方法和软件研制均有重要的价值.%The research of the three-dimensional compressible miscible (oil and water) displace-ment problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of three-dimensional bounded region, the authors put forward a kind of upwind finite difference schemes and make use the calculus of variations, the change of variables and the theory of a priori estimates and techniques. Optimal order estimates in l2 norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, model numerical method and software development. Thus, the well-known problem is solved.
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Kamynin, L I; Khimchenko, B N
2001-08-31
We consider two classes of second-order parabolic matrix-vector systems (with solutions u element of M{sub mx1}, m{>=}2) that can be reduced to a single second-order parabolic equation for a scalar function v=
, where p element of M{sub mx1} is a fixed stochastic constant vector. We consider the first boundary-value problem for a scalar second-order parabolic equation (with unbounded coefficients) in a domain unbounded with respect to x under the assumption of strong absorption at infinity. We obtain an a priori estimate for solutions of the first boundary-value problem in the generalized Tikhonov-Taecklind classes. (The problem under investigation has at most one solution in these classes.)
Mohd Fauzi, Norizyan Izzati; Sulaiman, Jumat
2013-04-01
The aim of this paper is to describe the application of Quarter-Sweep Gauss-Seidel (QSGS) iterative method using quadratic spline scheme for solving fourth order two-point linear boundary value problems. In the line to derive approximation equations, firstly the fourth order problems need to be reduced onto a system of second-order two-point boundary value problems. Then two linear systems have been constructed via discretization process by using the corresponding quarter-sweep quadratic spline approximation equations. The generated linear systems have been solved using the proposed QSGS iterative method to show the superiority over Full-Sweep Gauss-Seidel (FSGS) and Half-Sweep Gauss-Seidel (HSGS) methods. Computational results are provided to illustrate that the effectiveness of the proposed QSGS method is more superior in terms of computational time and number of iterations as compared to other tested methods.
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Alfonso Castro
1998-01-01
Full Text Available In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is $+1$. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i the existence of at least four solutions (one of which changes sign exactly once, ii the existence of at least five solutions (two of which change sign, and iii the existence of precisely two sign-changing solutions.
Nonlinear predator-prey singularly perturbed Robin Problems for reaction diffusion systems
Institute of Scientific and Technical Information of China (English)
莫嘉琪; 韩祥临
2003-01-01
The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.
Guibout, Vincent M.
This dissertation has been motivated by the need for new methods to address complex problems that arise in spacecraft formation design. As a direct result of this motivation, a general methodology for solving two-point boundary value problems for Hamiltonian systems has been found. Using the Hamilton-Jacobi theory in conjunction with the canonical transformation induced by the phase flow, it is shown that generating functions solve two-point boundary value problems. Traditional techniques for addressing these problems are iterative and require an initial guess. The method presented in this dissertation solves boundary value problems at the cost of a single function evaluation, although it requires knowledge of at least one generating function. Properties of this method are presented. Specifically, we show that it includes perturbation theory and generalizes it to nonlinear systems. Most importantly, it predicts the existence of multiple solutions and allows one to recover all of these solutions. To demonstrate the efficiency of this approach, an algorithm for computing the generating functions is proposed and its convergence properties are studied. As the method developed in this work is based on the Hamiltonian structure of the problem, particular attention must be paid to the numerics of the algorithm. To address this, a general framework for studying the discretization of certain dynamical systems is developed. This framework generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent and cotangent bundles respectively. In addition, it provides new insights into some symplectic integrators and leads to a new discrete Hamilton-Jacobi theory. Most importantly, it allows one to discretize optimal control problems. In particular, a discrete maximum principle is presented. This dissertation also investigates applications of the proposed method to solve two-point boundary value problems. In particular, new techniques for designing
A Third-Order p-Laplacian Boundary Value Problem Solved by an SL(3,ℝ Lie-Group Shooting Method
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Chein-Shan Liu
2013-01-01
Full Text Available The boundary layer problem for power-law fluid can be recast to a third-order p-Laplacian boundary value problem (BVP. In this paper, we transform the third-order p-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of r∈[0,1]. The present SL(3,ℝ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order p-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4 method to obtain a quite accurate numerical solution of the p-Laplacian.
Institute of Scientific and Technical Information of China (English)
沈春; 孙梅娜
2005-01-01
Based on the theory of stratification, the well-posedness of the initial and boundary value problems for the system of two-dimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
. The singularly perturbed Robin boundary value problems for the semilinear elliptic equation are considered.Under suitable conditions and by using the fixed point theorem the existence,uniqueness and asymptotic behavior of solution for the boundary value problems are studied.Received:2000-10-07.
Institute of Scientific and Technical Information of China (English)
Pankaj Kumar SRIVASTAVA; Manoj KUMAR
2012-01-01
A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows.The authors show that the approximate solutions obtained by the numerical algorithm developed by using nonpolynomial quintic spline functions are better than those produced by other spline and domain decomposition methods.The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.
Institute of Scientific and Technical Information of China (English)
崔国忠; 张志平; 江成顺
2002-01-01
This paper deals with the initial boundary value problem for the BoltzmannPoisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.
一阶混合型方程组的间断边值问题%DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR FIRSTORDER SYSTEMS OF MIXED TYPE
Institute of Scientific and Technical Information of China (English)
闻国椿; 杨丕文
2001-01-01
This paper deals with discontinuous boundary value problems for first order linear systems of mixed (elliptic-hyperbolic) type equations in a simply connected domain.Firstly, we derive the representation of solutions of the above boundary value problems for the mixed system, and then by using the method of successive iteration, the existence and uniqueness of solutions for the above problems are proved. From these results, we can derive the result on the Tricomi problem for Lavrent'ev-Bitsadze equation uxx+sgnyuyy=0 obtained by A.V.Bitsadze.%本文论讨单连通区域上一阶线性混合型(椭圆—双曲型)方程组的间断边值问题.我们首先给出混合型方程组特别是最简单的混合型方程组解的表示式，然后使用逐次逼近法，证明上述边值问题解的存在性与唯一性.由以上结果，可导出A.V.Bitsadze所得的Lavrent’ev-Bitsadze方程：uxx+sgnyuyy=0的Tricomi问题的可解性.
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George L. Karakostas
2006-08-01
Full Text Available We provide sufficient conditions for the existence of positive solutions of a three-point boundary value problem concerning a second order delay differential equation with damping and forcing term whose the delayed part is an actively bounded function, a meaning introduced in [19]. By writing the damping term as a difference of two factors one can extract more information on the solutions. (For instance, in an application, given in the last section, we can give the exact value of the norm of the solution.
A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
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S. Djezzar
2006-02-01
Full Text Available We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.
An inverse problem of determining a nonlinear term in an ordinary differential equation
Kamimura, Yutaka
1998-01-01
An inverse problem for a nonlinear ordinary differential equation is discussed. We prove an existence theorem of a nonlinear term with which a boundary value problem admits a solution. This is an improvement of earlier work by A. Lorenzi. We also prove a uniqueness theorem of the nonlinear term.
A priori estimates for the existence of a solution for a multi-point boundary value problem
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Trofimchuk Sergei
2000-01-01
Full Text Available Let , , , , with be given. Let be such that , be given. This paper is concerned with the problem of obtaining Poincaré type a priori estimates of the form . The study of such estimates is motivated by the problem of existence of a solution for the Caratheodory equation , , satisfying boundary conditions . This problem was studied earlier by Gupta et al. (Jour. Math. Anal. Appl. 189 (1995, 575–584 when the 's, all had the same sign.
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Pleshchinskaya, I.E.; Pleshchinskii, N.B.
1995-05-20
In this paper the authors study linear systems of first-order partial differential equations u{sub x} + Au{sub y} + Bu = O, where A and B are given constant square matrices and u is an unknown vector-valued function with values in an m-dimensional real or complex vector space. We exhibit cases in which the general solution of the system (1) can be expressed in terms of solutions of coupled second-order partial differential equations and the derivations of these solutions. Then the solution of boundary-value problems for the system (1) with a boundary condition of the form u = f reduces to the successive solution of boundary-value problems of the same type for such equations. 1{degree}. Consider the simplest special case of the system u{sub x} + Au{sub y} = O. We shall assume that A is a complex-valued matrix and u a complex-vector-valued function.
Eremeyev, Victor A.; Lebedev, Leonid P.
2016-03-01
Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin-Murdoch model to the Steigmann-Ogden model of surface elasticity using techniques from the theory of Sobolev's spaces and methods of functional analysis. The Steigmann-Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin-Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin-Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann-Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.
On a character of the forced vibrations of two-layer plate in the second boundary value problem
Directory of Open Access Journals (Sweden)
Poghosyan H. M.
2007-06-01
Full Text Available The three-dimensional dynamic problem of the elasticity theory on forced vibration of orthotropic plate at coulomb friction between layers is solved by the asymptotic method. The bottom obverse surface is subject to external dynamic influences, and top - is rigidly fixed. The common asymptotic solution of the problem is found. The closed solution for particular type of problems is found. The resonance arising conditions are established. It is known, that constant tangential displacements acting to the second layer do not influence in stress-strain state of the first layer. It is shown, that the same phenomenon with the big accuracy remains in force at linearly varying on coordinates influences.
一类四阶边值问题的变号解%Sign-changing Solutions to Fourth-order Boundary Value Problem
Institute of Scientific and Technical Information of China (English)
李宇华
2011-01-01
The problem U(4) (t) = f (t ,U) , t ∈ (o, 1) with the boundary value conditions u″ (0) = u″ (1) = O ,u (O)=u(l) =O is studied by using topological degree and Morse theory. Under some conditions, we obtain this problem has at least six solutions,including two positive solutions, two negative solutions and two sign-changing solutions.%利用拓扑度理论和Morse理论研究方程u(4)(t)=f(t,u),t∈(0,1),且带有边界条件u"(0)=u"(1)=0,u(0)=u(1)=0.在一定条件下,得到此问题有六个解,其中两个正解,两个负解,两个变号解.
Vdovichenko, I. I.; Yakovlev, M. Ya; Vershinin, A. V.; Levin, V. A.
2016-11-01
One of the key problems of mechanics of composite materials is an estimation of effective properties of composite materials. This article describes the algorithms for numerical evaluation of the effective thermal conductivity and thermal expansion of composites. An algorithm of effective thermal conductivity evaluation is based on sequential solution of boundary problems of thermal conductivity with different boundary conditions (in the form of the temperature on the boundary) on representative volume element (RVE) of composite with subsequent averaging of the resulting vector field of heat flux. An algorithm of effective thermal expansion evaluation is based on the solution of the boundary problem of elasticity (considering the thermal expansion) on a RVE of composite material with subsequent averaging of a resulting strain tensor field. Numerical calculations were performed with the help of Fidesys Composite software module of CAE Fidesys using the finite element method. The article presents the results of numerical calculations of the effective coefficients of thermal conductivity and thermoelasticity for two types of composites (single-layer fiber and particulate materials) in comparison with the analytical estimates. The comparison leads to the conclusion about the correctness of algorithms and program developed.
Energy Technology Data Exchange (ETDEWEB)
Vdovin, V.; Watari, T. [National Inst. for Fusion Science, Nagoya (Japan); Fukuyama, A.
1997-12-31
In the work we formulate the basic equations to solve the above ICRF problem in flux coordinates on different equilibria. The kinetic effects like cyclotron and Cherenkov absorptions, along with excitation of kinetic Alfven waves and finite Larmor radius effects are included. The ICRF plasma heating ({omega} {approx} {omega}{sub ci}) methods are prepared for the newly constructed LHD and projected W7-X stellarators or are conducted on operating machines like W7-AS, CHS, etc. For their adequate ICRH modelling and antenna development it is needed to create more complicated in compare with tokamaks ICRF code accounting for non axis symmetrical plasmas in complicated geometry. (author)