Bessel's Differential Equation and Its Hyers-Ulam Stability
Kim Byungbae
2007-01-01
Full Text Available We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation.
Bessel's Differential Equation and Its Hyers-Ulam Stability
Soon-Mo Jung
2007-12-01
Full Text Available We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation.
Identities involving Bessel polynomials arising from linear differential equations
Kim, Taekyun; Kim, Dae San
2016-01-01
In this paper, we study linear di?erential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.
Conformable Fractional Bessel Equation and Bessel Functions
Gökdoğan, Ahmet; Ünal, Emrah; Çelik, Ercan
2015-01-01
In this work, we study the fractional power series solutions around regular singular point x=0 of conformable fractional Bessel differential equation and fractional Bessel functions. Then, we compare fractional solutions with ordinary solutions. In addition, we present certain property of fractional Bessel functions.
YADOLLAH ORDOKHANI; HANIYE DEHESTANI
2015-01-01
In this paper a collocation method based on the Bessel-hybrid functions is used for approximation of the solution of linear Fredholm-Volterra integro-differential equations (FVIDEs) under mixed conditions. First, we explain the properties of Bessel-hybrid functions, which are combination of block-pulse functions and Bessel functions of first kind. The method is based upon Bessel-hybrid approximations, so that to obtain the operational matrixes and approximation of functions we use the tran...
YADOLLAH ORDOKHANI
2015-12-01
Full Text Available In this paper a collocation method based on the Bessel-hybrid functions is used for approximation of the solution of linear Fredholm-Volterra integro-differential equations (FVIDEs under mixed conditions. First, we explain the properties of Bessel-hybrid functions, which are combination of block-pulse functions and Bessel functions of first kind. The method is based upon Bessel-hybrid approximations, so that to obtain the operational matrixes and approximation of functions we use the transfer matrix from Bessel-hybrid functions to Taylor polynomials. The matrix equations correspond to a system of linear algebraic equations with the unknown Bessel-hybrid coefficients. Present results and comparisons demonstrate our estimate have good degree of accuracy.
The Bessel polynomials and their differential operators
Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs
FAN Hong-Yi; WANG Yong
2006-01-01
With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.
Time-reversal and the Bessel equation
Alfinito, Eleonora; Vitiello, Giuseppe
2015-07-01
The system of two damped/amplified oscillator equations is of widespread interest in the study of many physical problems and phenomena, from inflationary models of the Universe to thermal field theories, in condensed matter physics as well in high energy physics, and also in neuroscience. In this report we review the equivalence, in a suitable parametrization, between such a system of equations and the Bessel equations. In this connection, we discuss the breakdown of loop-antiloop symmetry, its relation with time-reversal symmetry and the mechanism of group contraction. Euclidean algebras such as e(2) and e(3) are also discussed in relation with Virasoro-like algebra.
Bessel functions and equations of mathematical physics
Epelde García, Markel
2016-01-01
The aim of this dissertation is to introduce Bessel functions to the reader, as well as studying some of their properties. Moreover, the final goal of this document is to present the most well- known applications of Bessel functions in physics.
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
Fractional Solutions of Bessel Equation with N-Method
Erdal Bas; Resat Yilmazer; Etibar Panakhov
2013-01-01
This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N ν method, we derive the fractional solutions of the equation.
Tricomi, FG
2012-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential
Hochstadt, Harry
2012-01-01
Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.
Bessel equation as an operator identity's matrix element in quantum mechanics
We study the well-known Bessel equation itself in the framework of quantum mechanics. We show that the Bessel equation is a spontaneous result of an operator identity's matrix element in some definite entangled state representations, which is a fresh look. Application of this operator formalism in the Hankel transform of Laplace equation is presented
Ultradiscrete limit of Bessel function type solutions of the Painlev\\'{e} III equation
Isojima, Shin
2014-01-01
An ultradiscrete analog of the Bessel function is constructed by taking the ultradiscrete limit for a $q$-difference analog of the Bessel function. Then, a direct relationship between a class of special solutions for the ultradiscrete Painlev\\'{e} III equation and those of the discrete Painlev\\'{e} III equation which have a determinantal structure is established.
McDonald, Kirk T
2000-01-01
Scalar Bessel beams are derived both via the wave equation and via diffraction theory. While such beams have a group velocity that exceeds the speed of light, this is a manifestation of the "scissors paradox" of special relativty. The signal velocity of a modulated Bessel beam is less than the speed of light. Forms of Bessel beams that satisfy Maxwell's equations are also given.
On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind
Dinesh Kumar
2015-01-01
Full Text Available We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably new results.
One some differential subordination involving the Bessel-Struve kernel function
Saiful R. Mondal; Dhuian, Mohamed Al
2016-01-01
In this article we study the inclusion properties of the Bessel-Struve kernel functions in the Janowski class. In particular, we find the conditions for which the Bessel-Struve kernel functions maps the unit disk to right half plane. An open problem in this aspect are also given. The third order differential subordination involving the Bessel-Struve kernel is also considered. The results are derived by defining suitable classes of admissible functions. One of the recurrence relation of the Be...
Mehmet Zeki SARIKAYA; Hüseyin YILDIRIM
2009-01-01
In this paper, we define the generalized Gauss-Weierstrass semigroups with Weierstrass kernel, and give some of their properties. Using them, we study the inversion formulas for the general-ized Riesz and Bessel potentials, generated by the generalized shift operators and associated with the Laplace Bessel differential operator.
Giuseppe Dattoli
1996-05-01
Full Text Available q analog of bessel functions, symmetric under the interchange of q and q^ −1 are introduced. The definition is based on the generating function realized as product of symmetric q-exponential functions with appropriate arguments. Symmetric q-Bessel function are shown to satisfy various identities as well as second-order q-differential equations, which in the limit q → 1 reproduce those obeyed by the usual cylindrical Bessel functions. A brief discussion on the possible algebraic setting for symmetric q-Bessel functions is also provided.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
On hitting times, Bessel bridges and Schrödinger’s equation
Hernandez-del-Valle, Gerardo
2013-01-01
In this paper we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schr\\"odinger's equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov's theorem, which suggests a strategy to extend our ideas to problems in $\\mathbb{R}^n$ and general diffusions. This approach also leads to (c) because we may relate, through a Feynman-Kac represe...
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Ladder operators and recursion relations for the associated Bessel polynomials
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways
Ladder operators and recursion relations for the associated Bessel polynomials
Fakhri, H.; Chenaghlou, A.
2006-10-01
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Ladder operators and recursion relations for the associated Bessel polynomials
Fakhri, H. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Department of Theoretical Physics and Astrophysics, Physics Faculty, Tabriz University, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of)]. E-mail: hfakhri@ipm.ir; Chenaghlou, A. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Physics Department, Faculty of Science, Sahand University of Technology, PO Box 51335-1996, Tabriz (Iran, Islamic Republic of)]. E-mail: a.chenaghlou@sut.ac.ir
2006-10-30
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
Solving Ordinary Differential Equations
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Nonlinear differential equations
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
Modern introduction to differential equations
Ricardo, Henry J
2009-01-01
A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equat
Elliptic partial differential equations
Volpert, Vitaly
If we had to formulate in one sentence what this book is about it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Mathematical anaylsis of reaction-diffusion equations will be based on the theory of Fredholm operators presented in the first volume. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equ...
On the Operator ⨁Bk Related to Bessel Heat Equation
Wanchak Satsanit
2010-01-01
Full Text Available We study the equation (∂/∂tu(x,t=c2⊕Bku(x,t with the initial condition u(x,0=f(x for x∈Rn+. The operator ⊕Bk is the operator iterated k-times and is defined by ⊕Bk=((∑i=1pBxi4-(∑j=p+1p+qBxi4k, where p+q=n is the dimension of the Rn+, Bxi=∂2/∂xi2+(2vi/xi(∂/∂xi, 2vi=2αi+1, αi>-1/2, i=1,2,3,…,n, and k is a nonnegative integer, u(x,t is an unknown function for (x,t=(x1,x2,…,xn,t∈Rn+×(0,∞, f(x is a given generalized function, and c is a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
赵喜来; 崔群法
2009-01-01
Root of characteristic equation for cylindrical Bessel equation eigenvalue prob-lems on general interval is of great real physical importance at engineering and physical. First, the characteristic equation of cylindrical Bessel equation eigenvalue problem on general interval is given, second, by mean of compared method, we obtaining roots of characteristic equation with Matlab program is discussed.
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
Differential Equations as Actions
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
On rough differential equations
Lejay, Antoine
2009-01-01
We prove that the Ito map, that is the map that gives the solution of a differential equation controlled by a rough path of finite p-variation with p in [2,3) is locally Lipschitz continuous in all its arguments and could be extended to vector fields that have only a linear growth.
Modelling by Differential Equations
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
Do Differential Equations Swing?
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Stochastic differential equations and applications
Friedman, Avner
2006-01-01
This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic es
Approximation of Analytic Functions by Bessel's Functions of Fractional Order
Soon-Mo Jung
2011-01-01
Full Text Available We will solve the inhomogeneous Bessel's differential equation x2y″(x+xy′(x+(x2-ν2y(x=∑m=0∞amxm, where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.
Approximation of Analytic Functions by Bessel's Functions of Fractional Order
Soon-Mo Jung
2011-01-01
We will solve the inhomogeneous Bessel's differential equation x2y″(x)+xy′(x)+(x2-ν2)y(x)=∑m=0∞amxm, where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Anticipated backward stochastic differential equations
Peng, Shige; Yang, Zhe
2009-01-01
In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.
Awojoyogbe, Bamidele O; Dada, Michael O; Onwu, Samuel O; Ige, Taofeeq A; Akinwande, Ninuola I
2016-04-01
Magnetic resonance imaging (MRI) uses a powerful magnetic field along with radio waves and a computer to produce highly detailed "slice-by-slice" pictures of virtually all internal structures of matter. The results enable physicians to examine parts of the body in minute detail and identify diseases in ways that are not possible with other techniques. For example, MRI is one of the few imaging tools that can see through bones, making it an excellent tool for examining the brain and other soft tissues. Pulsed-field gradient experiments provide a straightforward means of obtaining information on the translational motion of nuclear spins. However, the interpretation of the data is complicated by the effects of restricting geometries as in the case of most cancerous tissues and the mathematical concept required to account for this becomes very difficult. Most diffusion magnetic resonance techniques are based on the Stejskal-Tanner formulation usually derived from the Bloch-Torrey partial differential equation by including additional terms to accommodate the diffusion effect. Despite the early success of this technique, it has been shown that it has important limitations, the most of which occurs when there is orientation heterogeneity of the fibers in the voxel of interest (VOI). Overcoming this difficulty requires the specification of diffusion coefficients as function of spatial coordinate(s) and such a phenomenon is an indication of non-uniform compartmental conditions which can be analyzed accurately by solving the time-dependent Bloch NMR flow equation analytically. In this study, a mathematical formulation of magnetic resonance flow sequence in restricted geometry is developed based on a general second order partial differential equation derived directly from the fundamental Bloch NMR flow equations. The NMR signal is obtained completely in terms of NMR experimental parameters. The process is described based on Bessel functions and properties that can make it
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Khai, D. Q.; Tri, N. M.
2016-01-01
In this paper, we prove some results on the existence and decay properties of high order derivatives in time and space variables for local and global solutions of the Cauchy problem for the Navier-Stokes equations in Bessel-potential spaces.
Converting fractional differential equations into partial differential equations
He Ji-Huan; Li Zheng-Biao
2012-01-01
A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension.
Nambudiripad, K B M
2014-01-01
After presenting the theory in engineers' language without the unfriendly abstraction of pure mathematics, several illustrative examples are discussed in great detail to see how the various functions of the Bessel family enter into the solution of technically important problems. Axisymmetric vibrations of a circular membrane, oscillations of a uniform chain, heat transfer in circular fins, buckling of columns of varying cross-section, vibrations of a circular plate and current density in a conductor of circular cross-section are considered. The problems are formulated purely from physical considerations (using, for example, Newton's law of motion, Fourier's law of heat conduction electromagnetic field equations, etc.) Infinite series expansions, recurrence relations, manipulation of expressions involving Bessel functions, orthogonality and expansion in Fourier-Bessel series are also covered in some detail. Some important topics such as asymptotic expansions, generating function and Sturm-Lioville theory are r...
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Lopez, Cesar
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Differential Equations introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduct
Difference equations by differential equation methods
Hydon, Peter E
2014-01-01
Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.
Elements of partial differential equations
Sneddon, Ian N
2006-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Perturbed linear rough differential equations
Coutin, Laure; Lejay, Antoine
2014-01-01
We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Mag...
Scaling of differential equations
Langtangen, Hans Petter
2016-01-01
The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...
Some differential equations in synthetic differential geometry
Kock, Anders; Reyes, Gonzalo E.
2001-01-01
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations.
王强; 李顺初; 蒲俊
2015-01-01
This paper solved a boundary value problem of Riccati - Bessel equation;and the similar kernel function and similar structure of the solution were obtained. By further analysis and solving this class of boundary value problem,the guiding functions were firstly constructed by using two linearly independent solutions of Riccati- Bessel equation,and then the similar kernel function was assembled by the guiding functions and coefficient of right boundary value condition. The solution to the boundary value problem was assembled by similar kernel func-tion and coefficient of left boundary value condition. Therefore a new idea is put forward for solving this class of boundary value problem of Riccati - Bessel equation:similar structure.%针对 Riccati - Bessel 方程一类边值问题进行求解，获得了解式的相似核函数和相似结构，通过进一步分析，发现求解该类边值问题可先利用 Riccati - Bessel 方程的两个线性无关解构造引解函数，再结合右边值条件的系数组装得到相似核函数；通过相似核函数和左边值条件的系数组装就可以得到 Riccati - Bessel 方程边值问题的解，由此提出了解决该类 Riccati - Bessel 方程边值问题的一种新思路———相似构造。
Differential Equations for Algebraic Functions
Bostan, Alin; Chyzak, Frédéric; Salvy, Bruno; Lecerf, Grégoire; Schost, Éric
2007-01-01
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove ...
Introduction to ordinary differential equations
Rabenstein, Albert L
1966-01-01
Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations. This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutio
Introduction to partial differential equations from Fourier series to boundary-value problems
Broman, Arne
2010-01-01
This well-written, advanced-level text introduces students to Fourier analysis and some of its applications. The self-contained treatment covers Fourier series, orthogonal systems, Fourier and Laplace transforms, Bessel functions, and partial differential equations of the first and second orders. Over 260 exercises with solutions reinforce students' grasp of the material. 1970 edition.
Chaos in Partial Differential Equations
Li, Y. Charles
2009-01-01
This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n greater than 1). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each category, we pick a representative to present the results. Then we review some initial results o...
Differential equations and moving frames
Abib, Odinette Renée
2006-01-01
The purpose of the paper is to study the relationship between differential equations, Pfaffian systems and geometric structures, via the method of moving frames of E.Cartan. We show a local structure theorem. The Lie algebra aspects differential equations is studied too.
Solving Nonlinear Coupled Differential Equations
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Stochastic Geometric Partial Differential Equations
Brzezniak, Z.; Goldys, B.; Ondreját, Martin
1. Singapore : World Scientific Publishing Company, 2011 - (Zhao, H.; Truman, A.), s. 1-32 ISBN 978-981-4360-91-3. - (Interdisciplinary Mathematical Sciences. 12) R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic geometric * partial differential equations Subject RIV: BA - General Mathematics http://library.utia.cas.cz/separaty/2012/SI/ondrejat-stochastic geometric partial differential equations. pdf
Differential equations a concise course
Bear, H S
2011-01-01
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
Elliptic differential equations
Hackbusch, Wolfgang; Ion, PDF
2010-01-01
The book offers a simultaneous presentation of the theory and of the numerical treatment of elliptic problems. The author starts with a discussion of the Laplace equation in the classical formulation and its discretisation by finite differences and deals with topics of gradually increasing complexity in the following chapters. He introduces the variational formulation of boundary value problems together with the necessary background from functional analysis and describes the finite element method including the most important error estimates. A more advanced chapter leads the reader into the th
Exponential generating functions for the associated Bessel functions
Similar to the associated Legendre functions, the differential equation for the associated Bessel functions Bl,m(x) is introduced so that its form remains invariant under the transformation l → -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l l,m(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences
Introductory course on differential equations
Gorain, Ganesh C
2014-01-01
Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.
Group analysis of differential equations
Ovsiannikov, L V
1982-01-01
Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g
Differential equations and mathematical biology
Jones, DS; Sleeman, BD
2009-01-01
""… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.""-SIAM Review, Sept. 2010, Vol. 52, No. 3""… Where this text stands out i
Basic linear partial differential equations
Treves, Francois
2006-01-01
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their
Lectures on ordinary differential equations
Hurewicz, Witold
2014-01-01
Hailed by The American Mathematical Monthly as ""a rigorous and lively introduction,"" this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown funct
Loop equations from differential systems
Eynard, Bertrand; Marchal, Olivier
2016-01-01
To any differential system $d\\Psi=\\Phi\\Psi$ where $\\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\\Phi$ is a Lie algebra $\\mathfrak g$ valued 1-form on a Riemann surface $\\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.
Johannisson, P; Lisak, M; Marklund, M; Johannisson, Pontus; Anderson, Dan; Lisak, Mietek; Marklund, Mattias
2003-01-01
The effect of the Kerr nonlinearity on linear non-diffractive Bessel beams is investigated analytically and numerically using the nonlinear Schr\\"odinger equation. The nonlinearity is shown to primarily affect the central parts of the Bessel beam, giving rise to radial compression or decompression depending on whether the nonlinearity is focusing or defocusing, respectively. The dynamical properties of Gaussian-truncated Bessel beams are also analysed in the presence of a Kerr nonlinearity. It is found that although a condition for width balance in the root-mean-square sense exists, the beam profile becomes strongly deformed during propagation and may exhibit the phenomena of global and partial collapse.
Introduction to Piecewise Differentiable Equations
Scholtes, Stefan
2012-01-01
This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the non smooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop
Convexity, Differential Equations, and Games
Flåm, Sjur Didrik
2002-01-01
Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Stability of Functional Differential Equations
Lemm, Jeffrey M
1986-01-01
This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.
Groote, S; Pivovarov, A A
2012-01-01
We use configuration space methods to write down one-dimensional integral representations for one- and two-loop sunrise diagrams (also called Bessel moments) which we use to numerically check on the correctness of the second order differential equations for one- and two-loop sunrise diagrams that have recently been discussed in the literature.
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
Algebrization of Nonautonomous Differential Equations
María Aracelia Alcorta-García
2015-01-01
Full Text Available Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w, conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w=H(te,w and the maps H1(τ=H(τ,ξ and H2(ξ=H(τ,ξ are Lorch differentiable with respect to A for all (τ,ξ∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ of the differential equation dξ/dτ=H(τ,ξ over A define solutions (x(t,y(t=ξ(te of the planar system.
Theory of Bessel Functions of High Rank - I: Fundamental Bessel Functions
Qi, Zhi
2014-01-01
In this article we introduce a new category of special functions called fundamental Bessel functions arising from the Voronoi summation formula for $\\mathrm{GL}_n (\\mathbb{R})$. The fundamental Bessel functions of rank one and two are the oscillatory exponential functions $e^{\\pm i x}$ and the classical Bessel functions respectively. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel equations.
Partial differential equations an introduction
Colton, David
2004-01-01
Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
Stability theory of differential equations
Bellman, Richard
2008-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
Sensitivity of rough differential equations
Coutin, Laure; Lejay, Antoine
2015-01-01
In the context of rough paths theory, we study the regularity of the Itô map with respect to the starting point, the coefficients and perturbation of the driving rough paths. In particular, we show that the Itô mapis differentiable with a Hölder or Lipschitz continuousderivatives under general hypotheses. With respect to the current literatureon the subject, our proof relies on perturbation of linearRough Differential Equations, and the Hölder regularity of the Itô map is established.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
Handbook of differential equations stationary partial differential equations
Chipot, Michel
2006-01-01
This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Ke
Exterior Differential Systems for Higher Order Partial Differential Equations
Paul Bracken
2010-01-01
The theory of exterior differential systems is applied to study integrability of a set of related partial differential equations expressed in the terms of differential forms using Cartan's method. The Camassa-Holm equation and the Degasperis-Procesi equations are special cases that are described by these exterior differential systems. Some conservation laws are obtained in the cases of the more relevant equations. A closed differential ideal is constructed for each case studied.
Exterior Differential Systems for Higher Order Partial Differential Equations
Paul Bracken
2010-01-01
Full Text Available The theory of exterior differential systems is applied to study integrability of a set of related partial differential equations expressed in the terms of differential forms using Cartan's method. The Camassa-Holm equation and the Degasperis-Procesi equations are special cases that are described by these exterior differential systems. Some conservation laws are obtained in the cases of the more relevant equations. A closed differential ideal is constructed for each case studied.
Fractional Complex Transform for Fractional Differential Equations
Li, Zheng-Biao; He, Ji-Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.
Differential Equations of Ideal Memristors
Z. Biolek
2015-06-01
Full Text Available Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual characteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge, or Parameter-vs-state- map (PSM, mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM, composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and current is carried out.
Conservation Laws of Differential Equations in Finance
QIN Mao-Chang; MEI Feng-Xiang; SHANG Mei
2005-01-01
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Conservation Laws of Differential Equations in Finance
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Differential Galois Theory of Linear Difference Equations
Hardouin, Charlotte; Singer, Michael F.
2008-01-01
We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.
On Backward Stochstic Partial Differential Equations.
2001-01-01
We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations.
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Partial differential equations for scientists and engineers
Farlow, Stanley J
1993-01-01
Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing th
Partial differential equations of mathematical physics
Sobolev, S L
1964-01-01
Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied math
Introduction to differential equations with dynamical systems
Campbell, Stephen L
2011-01-01
Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Cam
Involutive reductions and solutions of differential equations
Engelmann, Joachim
2003-01-01
This work introduces the so-called involutive reduction procedure to simplify and solve differential equations. The method is based on symmetry analysis, which was developed by the Norwegian mathematician Sophus Lie. The involutive reduction procedure uses the given differential equation itself and the invariant surface condition of this differential equation. The invariant surface condition incorporates the symmetries of the problem. This coupled system of partial differential equati...
An introduction to differential equations
Ladde, Anil G
2012-01-01
This is a twenty-first century book designed to meet the challenges of understanding and solving interdisciplinary problems. The book creatively incorporates "cutting-edge" research ideas and techniques at the undergraduate level. The book also is a unique research resource for undergraduate/graduate students and interdisciplinary researchers. It emphasizes and exhibits the importance of conceptual understandings and its symbiotic relationship in the problem solving process. The book is proactive in preparing for the modeling of dynamic processes in various disciplines. It introduces a "break-down-the problem" type of approach in a way that creates "fun" and "excitement". The book presents many learning tools like "step-by-step procedures (critical thinking)", the concept of "math" being a language, applied examples from diverse fields, frequent recaps, flowcharts and exercises. Uniquely, this book introduces an innovative and unified method of solving nonlinear scalar differential equations. This is called ...
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Sobolev gradients and differential equations
Neuberger, John William
1997-01-01
A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.
Difference Galois theory of linear differential equations
Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael
2013-01-01
We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.
Differential difference inequalities related to parabolic functional differential equations
Milena Netka
2010-01-01
Full Text Available Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the method of lines is given. Nonlinear estimates of the Perron type for given operators with respect to functional variables are used. Results obtained in the paper can be applied to differential integral problems and to equations with deviated variables.
Ebaid, Abdelhalim; Al-Blowy, Ahmed B.
2016-05-01
In this article, a simple approach is suggested to calculate the approximate dates of opposition and conjunction of Earth and Mars since their opposition on August 28, 2003 (at perihelion of Mars). The goal of this article has been achieved via using accurate analytical solution to Kepler's equation in terms of Bessel function. The periodicity property of this solution and its particular values at specified times are discussed through some lemmas. The mathematical conditions of opposition and conjunction of the two planets are formulated. Moreover, the intervals of opposition and conjunction have been determined using the graphs of some defined functions. The calculations reveal that there are nine possible oppositions and conjunctions for Earth and Mars during 20 years started on August 28, 2003. The dates of such oppositions and conjunctions were approximately determined and listed in Tables. It is found that our calculations differ few days from the published real dates of Earth-Mars oppositions due to the neglected effects of the gravitational attraction of other planets in the Solar system on the motion of two planets. The period of 20 years can be extended for any number of years by following the suggested analysis. Furthermore, the current approach may be extended to study the opposition and conjunction of the Earth and any outer planet.
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Nth-order Fuzzy Differential Equations Under Generalized Differentiability
Soheil Salahshour
2011-11-01
Full Text Available In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space.
Nth-order Fuzzy Differential Equations Under Generalized Differentiability
Soheil Salahshour
2011-01-01
In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space.
A reinterpretation of set differential equations as differential equations in a Banach space
Rasmussen, Martin; Rieger, Janosch; Webster, Kevin
2015-01-01
Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example,...
Solutions manual to accompany Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Numerical methods for ordinary differential equations
Butcher, John C
2008-01-01
In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author''s pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book includeIntroductory work on differential and difference equations.A comprehensive introduction to the theory and practice of solving ordinary differential equations numeri...
Mathematical physics with partial differential equations
Kirkwood, James
2011-01-01
Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform
Solutions of fuzzy differential equations based on generalized differentiability
Bede , Barnabás; Gal, Sorin G.
2010-01-01
The main goal of this paper is to show that the concept of generalized differentiability introduced by the authors in [2] allows to obtain new solutions to the fuzzy differential equations. A very general existence and uniqueness result of two solutions for the fuzzy differential equations with modified argument and based on generalized differentiability is obtained together with a characterization of these solutions by ODEs. Real-world applications consisting in application...
A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations
Ivaz, K.; Khastan, A.; Juan J. Nieto
2013-01-01
Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples.
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Symbolic Solution of Linear Differential Equations
Feinberg, R. B.; Grooms, R. G.
1981-01-01
An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.
Solution techniques for elementary partial differential equations
Constanda, Christian
2012-01-01
Incorporating a number of enhancements, Solution Techniques for Elementary Partial Differential Equations, Second Edition presents some of the most important and widely used methods for solving partial differential equations (PDEs). The techniques covered include separation of variables, method of characteristics, eigenfunction expansion, Fourier and Laplace transformations, Green’s functions, perturbation methods, and asymptotic analysis.
Solving Differential Equations Using Modified Picard Iteration
Robin, W. A.
2010-01-01
Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…
Algebraic differential equations associated to some polynomials
Barlet, Daniel
2013-01-01
We compute the Gauss-Manin differential equation for any period of a polynomial in \\ $\\C[x_{0},\\dots, x_{n}]$ \\ with \\ $(n+2)$ \\ monomials. We give two general factorizations theorem in the algebra \\ $\\C$ \\ for such a differential equations.
ON ALGEBRICO-DIFFERENTIAL EQUATIONS-SOLVING
WU Wenjun(Wu Wen-tsun)
2004-01-01
The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations. As an illustration of the method, the Devil's Problem of Pommaret is solved in details.
Lakrib, Mustapha; Sari, Tewfik
2006-01-01
We prove averaging theorems for ordinary differential equations and retarded functional differential equations. Our assumptions are weaker than those required in the results of the existing literature. Usually, we require that the nonautonomous differential equation and the autonomous averaged equation are locally Lipschitz and that the solutions of both equations exist on some interval. We extend this result to the case of vector fields which are continuous in the spatial variable uniformly ...
Stochastic differential equation model to Prendiville processes
Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Stochastic differential equation model to Prendiville processes
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution
Markov processes and parabolic partial differential equations
Bossy, Mireille; Champagnat, Nicolas
2010-01-01
In the first part of this article, we present the main tools and definitions of Markov processes' theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov's backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devoted to the links between Markov processes and parabolic partial differential equations (PDEs). In part...
Differential equations inverse and direct problems
Favini, Angelo
2006-01-01
DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA
Symmetry Coefficients of Semilinear Partial Differential Equations
Freire, Igor Leite; Martins, Antonio Carlos Gilli
2008-01-01
We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to enlighte...
The Numerical Approximation of Functional Differential Equations
Venturi, Daniele
2016-01-01
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equations), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective action methods). However, no effective numerical method has yet been developed to compute their solution. The purpose of this manuscript is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations. The proposed methods will be described and demonstrated in various examples.
ON A CORRELATION BETWEEN DIFFERENTIAL EQUATIONS AND THEIR CHARACTERISTIC EQUATIONS
Boro M. Piperevski
2007-01-01
Abstract: The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.
Exponential generating functions for the associated Bessel functions
Fakhri, H [Department of Theoretical Physics and Astrophysics, Physics Faculty, University of Tabriz, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of); Mojaveri, B; Nobary, M A Gomshi [Department of Physics, Faculty of Science, Razi University, Kermanshah (Iran, Islamic Republic of)], E-mail: hfakhri@tabrizu.ac.ir, E-mail: bmojaveri@raziu.ac.ir, E-mail: mnobary@razi.ac.ir
2008-09-26
Similar to the associated Legendre functions, the differential equation for the associated Bessel functions B{sub l,m}(x) is introduced so that its form remains invariant under the transformation l {yields} -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l < 0 and l {>=} 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l - m) but with different values of l and m are independent of each other. So, all the functions B{sub l,m}(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.
Exponential generating functions for the associated Bessel functions
Fakhri, H.; Mojaveri, B.; Gomshi Nobary, M. A.
2008-09-01
Similar to the associated Legendre functions, the differential equation for the associated Bessel functions Bl,m(x) is introduced so that its form remains invariant under the transformation l → -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l = 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l - m) but with different values of l and m are independent of each other. So, all the functions Bl,m(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.
Partial Differential Equations Modeling and Numerical Simulation
Glowinski, Roland
2008-01-01
This book is dedicated to Olivier Pironneau. For more than 250 years partial differential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at first and then those originating from human activity and technological development. Mechanics, physics and their engineering applications were the first to benefit from the impact of partial differential equations on modeling and design, but a little less than a century ago the Schrödinger equation was the key opening the door to the application of partial differential equations to quantum chemistry, for small atomic and molecular systems at first, but then for systems of fast growing complexity. Mathematical modeling methods based on partial differential equations form an important part of contemporary science and are widely used in engineering and scientific applications. In this book several experts in this field present their latest results and discuss trends in the numerical analy...
Differential difference inequalities related to parabolic functional differential equations
Milena Netka
2010-01-01
Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the method of lines is given. Nonlinear estimates of the Perron type for given operators with respect to functional variables are used. Results obtained in the paper can be app...
Filamentation with nonlinear Bessel vortices.
Jukna, V; Milián, C; Xie, C; Itina, T; Dudley, J; Courvoisier, F; Couairon, A
2014-10-20
We present a new type of ring-shaped filaments featured by stationary nonlinear high-order Bessel solutions to the laser beam propagation equation. Two different regimes are identified by direct numerical simulations of the nonlinear propagation of axicon focused Gaussian beams carrying helicity in a Kerr medium with multiphoton absorption: the stable nonlinear propagation regime corresponds to a slow beam reshaping into one of the stationary nonlinear high-order Bessel solutions, called nonlinear Bessel vortices. The region of existence of nonlinear Bessel vortices is found semi-analytically. The influence of the Kerr nonlinearity and nonlinear losses on the beam shape is presented. Direct numerical simulations highlight the role of attractors played by nonlinear Bessel vortices in the stable propagation regime. Large input powers or small cone angles lead to the unstable propagation regime where nonlinear Bessel vortices break up into an helical multiple filament pattern or a more irregular structure. Nonlinear Bessel vortices are shown to be sufficiently intense to generate a ring-shaped filamentary ionized channel in the medium which is foreseen as opening the way to novel applications in laser material processing of transparent dielectrics. PMID:25401574
Symmetry analysis of differential equations an introduction
Arrigo, Daniel J
2015-01-01
A self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEsSymmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems. Thoroughly class-tested, t
Numerical Analysis of Partial Differential Equations
Lui, S H
2011-01-01
A balanced guide to the essential techniques for solving elliptic partial differential equations Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis
The Bessel Numbers and Bessel Matrices
Sheng Liang YANG; Zhan Ke QIAO
2011-01-01
In this paper,using exponential Riordan arrays,we investigate the Bessel numbers and Bessel matrices.By exploring links between the Bessel matrices,the Stirling matrices and the degenerate Stirling matrices,we show that the Bessel numbers are special case of the degenerate Stirling numbers,and derive explicit formulas for the Bessel numbers in terms of the Stirling numbers and binomial coefficients.
Sasaki, Ryu
2016-01-01
Exact solvability (ES) of one-dimensional quantum potentials $V(x)$ is a vague concept. We propose that beyond its most conventional range the ES status should be attributed also to many less common interaction models for which the wave functions remain piecewise proportional to special functions. The claim is supported by constructive analysis of a toy model $V(x)= -g^2\\exp (-|x|)$. The detailed description of the related bound-state and scattering solutions of Schr\\"{o}dinger equation is provided in terms of Bessel functions which are properly matched in the origin.
Trends in differential equations and applications
Neble, María; Galván, José
2016-01-01
This work collects the most important results presented at the Congress on Differential Equations and Applications/Congress on Applied Mathematics (CEDYA/CMA) in Cádiz (Spain) in 2015. It supports further research in differential equations, numerical analysis, mechanics, control and optimization. In particular, it helps readers gain an overview of specific problems of interest in the current mathematical research related to different branches of applied mathematics. This includes the analysis of nonlinear partial differential equations, exact solutions techniques for ordinary differential equations, numerical analysis and numerical simulation of some models arising in experimental sciences and engineering, control and optimization, and also trending topics on numerical linear Algebra, dynamical systems, and applied mathematics for Industry. This volume is mainly addressed to any researcher interested in the applications of mathematics, especially in any subject mentioned above. It may be also useful to PhD s...
Existence results for random fractional differential equations
Vasile Lupulescu; Donal O'Regan; Ghaus ur Rahman
2014-01-01
In this paper, an existence result for a random fractional differential equation is established under a Carathéodory condition. Existence results for extremal random solutions are also proved. Finally, an existence and uniqueness result is given
Fractional-differential equations of motion
Olemskoi, Alexander I.
1999-01-01
An elementary system leading to the notions of fractional integrals and derivatives is considered. Various physical situations whose description is associated with fractional differential equations of motion are discussed.
Connecting Related Rates and Differential Equations
Brandt, Keith
2012-01-01
This article points out a simple connection between related rates and differential equations. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.
Evaluating methods for approximating stochastic differential equations
Brown, Scott D; RATCLIFF, ROGER; Smith, Philip L.
2006-01-01
Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler’s method for solving ordinary differential equations. The accuracy of Euler’s method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matri...
Multidimensional stochastic differential equations with distributional drift
Flandoli, Franco; Issoglio, Elena; Russo, Francesco
2015-01-01
This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
A Unified Introduction to Ordinary Differential Equations
Lutzer, Carl V.
2006-01-01
This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)
Existence theorems for ordinary differential equations
Murray, Francis J
2007-01-01
Theorems stating the existence of an object-such as the solution to a problem or equation-are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations.The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, th
Some problems in fractal differential equations
Su, Weiyi
2016-06-01
Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.
5-th order differential equations related to Calabi-Yau differential equations
Almkvist, Gert
2007-01-01
The very few 5-th order differential equations which have 4-th order Calabi-Yau equations as pullbacks are listed. We use the pullback of Yifan Yang that in most cases has much lower degree than the usual pullback.
Differential geometry techniques for sets of nonlinear partial differential equations
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Modelling conjugation with stochastic differential equations
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik;
2010-01-01
Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...... using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared...
Fuzzy differential equations in various approaches
Gomes, Luciana Takata; Bede, Barnabas
2015-01-01
This book may be used as reference for graduate students interested in fuzzy differential equations and researchers working in fuzzy sets and systems, dynamical systems, uncertainty analysis, and applications of uncertain dynamical systems. Beginning with a historical overview and introduction to fundamental notions of fuzzy sets, including different possibilities of fuzzy differentiation and metric spaces, this book moves on to an overview of fuzzy calculus thorough exposition and comparison of different approaches. Innovative theories of fuzzy calculus and fuzzy differential equations using fuzzy bunches of functions are introduced and explored. Launching with a brief review of essential theories, this book investigates both well-known and novel approaches in this field; such as the Hukuhara differentiability and its generalizations as well as differential inclusions and Zadeh’s extension. Through a unique analysis, results of all these theories are examined and compared.
On the Diamond Bessel Heat Kernel
Wanchak Satsanit
2011-01-01
We study the heat equation in n dimensional by Diamond Bessel operator. We find the solution by method of convolution and Fourier transform in distribution theory and also obtain an interesting kernel related to the spectrum and the kernel which is called Bessel heat kernel.
Using Bessel functions to find fixed solutions to wave equations%利用Bessel函数求解波动方程的定解问题
金启胜
2014-01-01
分离变量法是求解波动方程定解问题的一种重要方法。分离变量法的重点在于求特征值及其对应的特征函数。Bessel函数是应用很广泛的一种特征函数，运用Bessel函数的有关性质可以很方便地求解波动方程的定解问题。%The method of separation of variables is an important method for finding solutions to heat conduction equations. The method is used to find the characteristic values of characteristic functions. Bessel functions are characteristic functions,which can be widely used to find fixed solutions to wave equations.
Monotonic sequences related to zeros of Bessel functions
Lorch, Lee; Muldoon, Martin
2008-12-01
In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J 0 at multiples of π, i.e., at the zeros of J 1/2. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.
Transform methods for solving partial differential equations
Duffy, Dean G
2004-01-01
Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach. Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements: New in the Second Edition: ·...
Inequalities for differential and integral equations
Ames, William F
1997-01-01
Inequalities for Differential and Integral Equations has long been needed; it contains material which is hard to find in other books. Written by a major contributor to the field, this comprehensive resource contains many inequalities which have only recently appeared in the literature and which can be used as powerful tools in the development of applications in the theory of new classes of differential and integral equations. For researchers working in this area, it will be a valuable source of reference and inspiration. It could also be used as the text for an advanced graduate course.Key Features* Covers a variety of linear and nonlinear inequalities which find widespread applications in the theory of various classes of differential and integral equations* Contains many inequalities which have only recently appeared in literature and cannot yet be found in other books* Provides a valuable reference to engineers and graduate students
A short course in ordinary differential equations
Kong, Qingkai
2014-01-01
This text is a rigorous treatment of the basic qualitative theory of ordinary differential equations, at the beginning graduate level. Designed as a flexible one-semester course but offering enough material for two semesters, A Short Course covers core topics such as initial value problems, linear differential equations, Lyapunov stability, dynamical systems and the Poincaré—Bendixson theorem, and bifurcation theory, and second-order topics including oscillation theory, boundary value problems, and Sturm—Liouville problems. The presentation is clear and easy-to-understand, with figures and copious examples illustrating the meaning of and motivation behind definitions, hypotheses, and general theorems. A thoughtfully conceived selection of exercises together with answers and hints reinforce the reader's understanding of the material. Prerequisites are limited to advanced calculus and the elementary theory of differential equations and linear algebra, making the text suitable for senior undergraduates as w...
A first course in differential equations
Logan, J David
2015-01-01
The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market. It presents a thorough treatment of the standard topics in an accessible, easy-to-read, format. The overarching perspective of the text conveys that differential equations are about applications. This book illuminates the mathematical theory in the text with a wide variety of applications that will appeal to students in physics, engineering, the biosciences, economics and mathematics. Instructors are likely to find that the first four or five chapters are suitable for a first course in the subject. This edition contains a healthy increase over earlier editions in the number of worked examples and exercises, particularly those routine in nature. Two appendices include a review with practice problems, and a MATLAB® supplement that gives basic codes and commands for solving differential equations. MATLAB® is not required; students are encouraged t...
Difference methods for stiff delay differential equations
Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = [y1(α1(y(t))),..., y/sub n/(α/sub n/(y(t)))]/sup T/ and α/sub i/(y(t)) less than or equal to t, arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solution of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. By use of the model equation y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, A( )-stability, and stiff stability have been generalize to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This straight forward extension is not true for implicit Runge-Kutta methods, as illustrated by the midpoint formula, which is A-stable for ordinary equations, but not for delay equations. A computer code for stiff delay equations was developed using the BDF. 24 figures, 5 tables
LINEARIZED OSCILLATIONS FOR NEUTRAL DIFFERENTIAL EQUATIONS
李建利; 申建华
2004-01-01
Consider the nonlinear neutral delay differential equation d/dt[x(t)-P(t)g(x(t-τ))]+Q(t)h(x(t-σ))=01t≥t0,where P1Q∈C([t01∞),R(τ>01,σ≥0,We obtain a linearized oscillation result by an associate linear equation in the case when liminf P(t)=p0∈[-1,0)。
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Eigenvalue problems for fractional ordinary differential equations
The eigenvalue problems are considered for the fractional ordinary differential equations with different classes of boundary conditions including the Dirichlet, Neumann, Robin boundary conditions and the periodic boundary condition. The eigenvalues and eigenfunctions are characterized in terms of the Mittag–Leffler functions. The eigenvalues of several specified boundary value problems are calculated by using MATLAB subroutine for the Mittag–Leffler functions. When the order is taken as the value 2, our results degenerate to the classical ones of the second-ordered differential equations. When the order α satisfies 1 < α < 2 the eigenvalues can be finitely many.
Surveys in differential-algebraic equations III
Reis, Timo
2015-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Flexibility of DAE formulations - Reachability analysis and deterministic global optimization - Numerical linear algebra methods - Boundary value problems The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
Surveys in differential-algebraic equations II
Reis, Timo
2015-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Observers for DAEs - DAEs in chemical processes - Optimal control of DAEs - DAEs from a functional-analytic viewpoint - Algebraic methods for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
A minicourse on stochastic partial differential equations
Rassoul-Agha, Firas
2009-01-01
In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time keeping a realistic pace for the student of the subject.
Algorithms For Integrating Nonlinear Differential Equations
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
Modeling and Prediction Using Stochastic Differential Equations
Juhl, Rune; Møller, Jan Kloppenborg; Jørgensen, John Bagterp;
2016-01-01
Pharmacokinetic/pharmakodynamic (PK/PD) modeling for a single subject is most often performed using nonlinear models based on deterministic ordinary differential equations (ODEs), and the variation between subjects in a population of subjects is described using a population (mixed effects) setup...... deterministic and can predict the future perfectly. A more realistic approach would be to allow for randomness in the model due to e.g., the model be too simple or errors in input. We describe a modeling and prediction setup which better reflects reality and suggests stochastic differential equations (SDEs) for...
Stochastic versus deterministic systems of differential equations
Ladde, G S
2003-01-01
This peerless reference/text unfurls a unified and systematic study of the two types of mathematical models of dynamic processes-stochastic and deterministic-as placed in the context of systems of stochastic differential equations. Using the tools of variational comparison, generalized variation of constants, and probability distribution as its methodological backbone, Stochastic Versus Deterministic Systems of Differential Equations addresses questions relating to the need for a stochastic mathematical model and the between-model contrast that arises in the absence of random disturbances/flu
Differential equations and applications recent advances
2014-01-01
Differential Equations and Applications : Recent Advances focus on the latest developments in Nonlinear Dynamical Systems, Neural Networks, Fluid Dynamics, Fractional Differential Systems, Mathematical Modelling and Qualitative Theory. Different aspects such as Existence, Stability, Controllability, Viscosity and Numerical Analysis for different systems have been discussed in this book. This book will be of great interest and use to researchers in Applied Mathematics, Engineering and Mathematical Physics.
Nonlinear partial differential equations on homogeneous spaces
The work of A.K.N.S. which is based on the s1(2,R) valued soliton connection is extended to obtain new integrable coupled nonlinear partial differential equations. This is achieved by assuming the soliton connection having values in a semisimple Lie, Kac-Moody, Lie super algebras. Extensions of some of the integrable nonlinear p.d.e's are given explicitly. In particular the coupled NLS equations on various homogeneous spaces and the coupled modified KdV equations are obtained on symmetric spaces. (author). 20 refs
Nonlinear Partial Functional Differential Equations: Existence and Stability
Caraballo Garrido, Tomás
2001-01-01
Existence and uniqueness of solutions for a class of nonlinear functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied. The stability results derived are also applied to ordinary differential equations with hereditary characteristics.
On fractional random differential equations with delay
Ho Vu; Nguyen Ngoc Phung; Nguyen Phuong
2016-01-01
In this paper, we consider the existence and uniqueness of solutions of the fractional random differential equations with delay. Moreover, some kind of boundedness of the solution is proven. Finally, the applicability of the theoretical results is illustrated with some real world examples.
Path Integral Methods for Stochastic Differential Equations
Chow, Carson C.; Buice, Michael A.
2015-01-01
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
Cartan normal conformal connections from differential equations
We explore and show a natural relationship between all third-order ordinary differential equations that possess a vanishing Wunschmann invariant, with conformal metrics on 3-manifolds and Cartan's normal O(3, 2) conformal connections. The generalization to pairs of second-order PDEs and their relationship to Cartan's normal O(4, 2) conformal connections on four-dimensional manifolds is discussed
Simplified differential equations approach for Master Integrals
Papadopoulos, Costas G.
2014-01-01
A simplified differential equations approach for Master Integrals is presented. It allows to express them, straightforwardly, in terms of Goncharov Polylogarithms. As a proof-of-concept of the proposed method, results at one and two loops are presented, including the massless one-loop pentagon with up to one off-shell leg at order epsilon.
Newton's method for stochastic functional differential equations
Monika Wrzosek
2012-08-01
Full Text Available In this article, we apply Newton's method to stochastic functional differential equations. The first part concerns a first-order convergence. We formulate a Gronwall-type inequality which plays an important role in the proof of the convergence theorem for the Newton method. In the second part a probabilistic second-order convergence is studied.
Stochastic differential equations used to model conjugation
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
Stochastic differential equations (SDEs) are used to model horizontal transfer of antibiotic resis- tance by conjugation. The model describes the concentration of donor, recipient, transconjugants and substrate. The strength of the SDE model over the traditional ODE models is that the noise can be...
Efficient Estimating Functions for Stochastic Differential Equations
Jakobsen, Nina Munkholt
The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over a...
Parallel Algorithm Solves Coupled Differential Equations
Hayashi, A.
1987-01-01
Numerical methods adapted to concurrent processing. Algorithm solves set of coupled partial differential equations by numerical integration. Adapted to run on hypercube computer, algorithm separates problem into smaller problems solved concurrently. Increase in computing speed with concurrent processing over that achievable with conventional sequential processing appreciable, especially for large problems.
Difference and differential equations with applications in queueing theory
Haghighi, Aliakbar Montazer
2013-01-01
A Useful Guide to the Interrelated Areas of Differential Equations, Difference Equations, and Queueing Models Difference and Differential Equations with Applications in Queueing Theory presents the unique connections between the methods and applications of differential equations, difference equations, and Markovian queues. Featuring a comprehensive collection of
Numerical methods for nonlinear partial differential equations
Bartels, Sören
2015-01-01
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.
Stochastic partial differential equations an introduction
Liu, Wei
2015-01-01
This book provides an introduction to the theory of stochastic partial differential equations (SPDEs) of evolutionary type. SPDEs are one of the main research directions in probability theory with several wide ranging applications. Many types of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. The theory of SPDEs is based both on the theory of deterministic partial differential equations, as well as on modern stochastic analysis. Whilst this volume mainly follows the ‘variational approach’, it also contains a short account on the ‘semigroup (or mild solution) approach’. In particular, the volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. Various types of generalized coercivity conditions are shown to guarantee non-explosion, but also a systematic approach to treat SPDEs with explosion in finite time is developed. It is, so far, the only book where the latter and t...
Numerical approximation of partial differential equations
Bartels, Sören
2016-01-01
Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular ...
Ordinary differential equations a graduate text
Bhamra, K S
2015-01-01
ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. The book discusses qualitative and quantitative aspects of the Strum - Liouville problems, Green's function, integral equations, Laplace transform and is supported by a number of worked-out examples in each lesson to make the concepts clear. A lot of stress on stability theory is laid down, especially on Lyapunov and Poincare stability theory. A numerous figures in various lessons (in particular lessons dealing with stability theory) have been added to clarify the key concepts in DE theory. Nonlinear oscillation in conservative systems and Hamiltonian systems highlights basic nature of the systems considered. Perturbation techniques lesson deals in fairly d...
陈宗荣; 李顺初
2011-01-01
In this paper, we solve the general boundary value problem of Bessel equations, and obtain the similar structure and the similar kernel function of solutions. A new idea and method for solving this kind of problems is formed: so called "similar structure construction". This idea and method can be used to analyze inner properties of solutions, to solve some practical problems and to write analytical softwares.%对Bessel方程的一般边值问题进行求解,得到了解式的相似结构和相似核函数及求解Bessel方程边值问题的一个新思想和新方法:相似结构构造法.该方法有利于进一步分析解的内在规律、解决相应的应用问题、方便编制相应的分析软件.
Differential constraints and exact solutions of nonlinear diffusion equations
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries
Differential constraints and exact solutions of nonlinear diffusion equations
Kaptsov, Oleg V.; Verevkin, Igor V.
2002-01-01
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries.
Introduction to numerical methods for time dependent differential equations
Kreiss, Heinz-Otto
2014-01-01
Introduces both the fundamentals of time dependent differential equations and their numerical solutions Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs). Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the t
Schiesser, William E
2014-01-01
Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the com
Schiesser, William E
2014-01-01
Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-worldODE problems that are found in a variety of fields, including chemistry, physics, biology,and physiology. The book provides readers with the necessary knowledge to reproduce andextend the comp
A textbook on ordinary differential equations
Ahmad, Shair
2014-01-01
The book is a primer of the theory of Ordinary Differential Equations. Each chapter is completed by a broad set of exercises; the reader will also find a set of solutions of selected exercises. The book contains many interesting examples as well (like the equations for the electric circuits, the pendium equation, the logistic equation, the Lotka-Volterra system, and many other) which introduce the reader to some interesting aspects of the theory and its applications. The work is mainly addressed to students of Mathematics, Physics, Engineering, Statistics, Computer Sciences, with knowledge of Calculus and Linear Algebra, and contains more advanced topics for further developments, such as Laplace transform; Stability theory and existence of solutions to Boundary Value problems. The authors are preparing a complete solutions manual, containing solutions to all the exercises published in the book. The manual will be available Summer 2014. Instructors who wish to adopt the book may request the manual by writing...
Renormalization methods for higher order differential equations
We adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second-order equation. In particular, we study equations involving the derivative of a double-well potential such as u − u3 or − u + 2u3. Using momentum (Fourier) space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation, once length is properly renormalized, yielding an approximation to solutions of the higher order equation from the second order. The results are confirmed using numerical computations. Additional numerics confirm that the main role of the higher order derivatives is in rescaling the length. (paper)
Malliavin differentiability of solutions of rough differential equations
Inahama, Yuzuru
2013-01-01
In this paper we study rough differential equations driven by Gaussian rough paths from the viewpoint of Malliavin calculus. Under mild assumptions on coefficient vector fields and underlying Gaussian processes, we prove that solutions at a fixed time is smooth in the sense of Malliavin calculus. Examples of Gaussian processes include fractional Brownian motion with Hurst parameter larger than $1/4$.
An introduction to ordinary differential equations
Coddington, Earl A
1989-01-01
""Written in an admirably cleancut and economical style."" - Mathematical Reviews. This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented. The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solu
ERC Workshop on Geometric Partial Differential Equations
Novaga, Matteo; Valdinoci, Enrico
2013-01-01
This book is the outcome of a conference held at the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. The aim of the conference was to discuss recent results on nonlinear partial differential equations, and more specifically geometric evolutions and reaction-diffusion equations. Particular attention was paid to self-similar solutions, such as solitons and travelling waves, asymptotic behaviour, formation of singularities and qualitative properties of solutions. These problems arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years.
Modification of Ordinary Differential Equations MATLAB Solver
E. Cocherova
2003-12-01
Full Text Available Various linear or nonlinear electronic circuits can be described bythe set of ordinary differential equations (ODEs. The ordinarydifferential equations can be solved in the MATLAB environment inanalytical (symbolic toolbox or numerical way. The set of nonlinearODEs with high complexity can be usually solved only by use ofnumerical integrator (solver. The modification of ode23 MATLABnumerical solver has been suggested in this article for the applicationin solution of some special cases of ODEs. The main feature of thismodification is that the solution is found at every prescribed point,in which the special behavior of system is anticipated. Theextrapolation of solution is not allowed in those points.
Causal interpretation of stochastic differential equations
Sokol, Alexander; Hansen, Niels Richard
2014-01-01
We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention...... structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a Lévy process, the postintervention distribution is identifiable from the generator of the SDE....
Jackson Integral Representations of Modified $q$-Bessel Functions and $q$-Bessel-Macdonald Functions
Olshanetsky, M. A.; Rogov, V-B. K.
1996-01-01
The $q$ analog of Modified Bessel functions and Bessel-Macdonald functions, were defined in our previous work (q-alg/950913) as general solutions of a second order difference equations. Here we present a collection of their representations by the Jackson q-integral.
Scattering of Bessel beam by arbitrarily shaped composite particles with core–shell structure
This study investigates the scattering of Bessel beam by composite particles with core–shell structure. Specifically, the vector expressions of zero-th order Bessel beam that satisfy well Maxwell's equations in combination with the rotation Euler angles are used to represent the arbitrarily incident Bessel beams. An efficient numerical method based on surface integral equations is introduced to formulate the scattering problems involving arbitrarily shaped composite particles with core–shell structure. Solutions are performed iteratively by using the multilevel fast multipole algorithm. The numerical results for differential scattering cross sections of several selected composite particles are presented and analyzed. This investigation is expected to provide useful guidance for techniques of laser detection on particle, diagnosis, and manipulation. - Highlights: • The scattering of Bessel beam by composite particles is numerically investigated. • A detailed description of the arbitrarily incident Bessel beam is presented. • The surface integral equation method is used to solve the studied scattering problem. • The numerical results of several typical composite particles are presented
Theory and applications of fractional differential equations
Kilbas, Anatoly A; Trujillo, Juan J; Van Mill, Jan
2006-01-01
This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus. The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Some of the areas of prese
Boundary differentiability for inhomogeneous infinity Laplace equations
Guanghao Hong
2014-03-01
Full Text Available We study the boundary regularity of the solutions to inhomogeneous infinity Laplace equations. We prove that if $u\\in C(\\bar{\\Omega}$ is a viscosity solution to $\\Delta_{\\infty}u:=\\sum_{i,j=1}^n u_{x_i}u_{x_j}u_{x_ix_j}=f$ with $f\\in C(\\Omega\\cap L^{\\infty}(\\Omega$ and for $x_0\\in \\partial\\Omega$ both $\\partial\\Omega$ and $g:=u|_{\\partial\\Omega}$ are differentiable at $x_0$, then u is differentiable at $x_0$.
Ordinary differential equations and mechanical systems
Awrejcewicz, Jan
2014-01-01
This book applies a step-by-step treatment of the current state-of-the-art of ordinary differential equations used in modeling of engineering systems/processes and beyond. It covers systematically ordered problems, beginning with first and second order ODEs, linear and higher-order ODEs of polynomial form, theory and criteria of similarity, modeling approaches, phase plane and phase space concepts, stability optimization, and ending on chaos and synchronization. Presenting both an overview of the theory of the introductory differential equations in the context of applicability and a systematic treatment of modeling of numerous engineering and physical problems through linear and non-linear ODEs, the volume is self-contained, yet serves both scientific and engineering interests. The presentation relies on a general treatment, analytical and numerical methods, concrete examples, and engineering intuition. The scientific background used is well balanced between elementary and advanced level, making it as a uniqu...
Hamiltonian partial differential equations and applications
Nicholls, David; Sulem, Catherine
2015-01-01
This book is a unique selection of work by world-class experts exploring the latest developments in Hamiltonian partial differential equations and their applications. Topics covered within are representative of the field’s wide scope, including KAM and normal form theories, perturbation and variational methods, integrable systems, stability of nonlinear solutions as well as applications to cosmology, fluid mechanics and water waves. The volume contains both surveys and original research papers and gives a concise overview of the above topics, with results ranging from mathematical modeling to rigorous analysis and numerical simulation. It will be of particular interest to graduate students as well as researchers in mathematics and physics, who wish to learn more about the powerful and elegant analytical techniques for Hamiltonian partial differential equations.
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Solving Partial Differential Equations Using a New Differential Evolution Algorithm
Natee Panagant
2014-01-01
Full Text Available This paper proposes an alternative meshless approach to solve partial differential equations (PDEs. With a global approximate function being defined, a partial differential equation problem is converted into an optimisation problem with equality constraints from PDE boundary conditions. An evolutionary algorithm (EA is employed to search for the optimum solution. For this approach, the most difficult task is the low convergence rate of EA which consequently results in poor PDE solution approximation. However, its attractiveness remains due to the nature of a soft computing technique in EA. The algorithm can be used to tackle almost any kind of optimisation problem with simple evolutionary operation, which means it is mathematically simpler to use. A new efficient differential evolution (DE is presented and used to solve a number of the partial differential equations. The results obtained are illustrated and compared with exact solutions. It is shown that the proposed method has a potential to be a future meshless tool provided that the search performance of EA is greatly enhanced.
On weak solutions of stochastic differential equations
Hofmanová, Martina; Seidler, Jan
2012-01-01
Roč. 30, č. 1 (2012), s. 100-121. ISSN 0736-2994 R&D Projects: GA ČR GAP201/10/0752 Institutional research plan: CEZ:AV0Z10750506 Keywords : stochastic differential equations * weak solutions Subject RIV: BA - General Mathematics Impact factor: 0.303, year: 2012 http://library.utia.cas.cz/separaty/2012/SI/hofmanova-0373626. pdf
Reducing differential equations for multiloop master integrals
Lee, Roman
2014-01-01
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degen...
Partial differential equation models in macroeconomics.
Achdou, Yves; Buera, Francisco J; Lasry, Jean-Michel; Lions, Pierre-Louis; Moll, Benjamin
2014-11-13
The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research. PMID:25288811
Spurious Solutions Of Nonlinear Differential Equations
Yee, H. C.; Sweby, P. K.; Griffiths, D. F.
1992-01-01
Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.
Observability of discretized partial differential equations
Cohn, Stephen E.; Dee, Dick P.
1988-01-01
It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.
Ambit processes and stochastic partial differential equations
Barndorff-Nielsen, Ole; Benth, Fred Espen; Veraart, Almut
Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection betwe...... ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis....
Spurious Numerical Solutions Of Differential Equations
Lafon, A.; Yee, H. C.
1995-01-01
Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.
Integrability Estimates for Gaussian Rough Differential Equations
Cass, Thomas; Lyons, Terry
2011-01-01
We derive explicit tail-estimates for the Jacobian of the solution flow of stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems.
Linear measure functional differential equations with infinite delay
Monteiro, G. (Giselle Antunes); Slavík, A.
2014-01-01
We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.
Teaching Modeling with Partial Differential Equations: Several Successful Approaches
Myers, Joseph; Trubatch, David; Winkel, Brian
2008-01-01
We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…
Synchronization with propagation - The functional differential equations
Rǎsvan, Vladimir
2016-06-01
The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators. On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems. The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.
Wilsonian renormalization, differential equations and Hopf algebras
Thomas, Krajewski
2008-01-01
In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Finally, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally...
A textbook on ordinary differential equations
Ahmad, Shair
2015-01-01
This book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, whic...
Complex Linearizability Criteria for Differential Equations
Ali, S; Qadir, Asghar
2010-01-01
The classical work of Lie for linearization of a scalar second-order ordinary differential equation (ODE) has recently been extended to higher order and to systems using real variables. There is little available of a general nature for partial differential equations (PDEs). Here we use complex functions of a complex variable to extend the earlier results to systems of PDEs and complex functions of a real variable to extend to systems of real variables. Linearization of nonlinear PDEs is one of the challenging problems in mathematics. There exists no invariant criteria to check the linearizability of such equations. In a non-trivial way, we gain explicit invariant linearizability conditions for a class of systems of non-linear PDEs which were not reported before. The problem of construction of linearizing transformation is also addressed which give way to procedures for writing down the solutions of the linearized equations. Besides, the integrability of systems of ODEs is achieved by taking advantage of analy...
NUMERICAL METHODS FOR DIFFERENTIAL GAMES BASED ON PARTIAL DIFFERENTIAL EQUATIONS
Falcone, M
2006-01-01
In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dyna...
Babusci, D.; Dattoli, G.; Germano, B.; Martinelli, M. R.; Ricci, P. E.
2011-01-01
We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.
Noncommutative Bessel symmetric functions
Novelli, Jean-Christophe; Thibon, Jean-Yves
2006-01-01
The consideration of tensor products of 0-Hecke algebra modules leads to natural analogs of the Bessel J-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.
Preconditioned iterative methods for partial differential equations
Leaf, G.K.; Minkoff, M.; Diaz, J.C.
1987-01-01
In this paper we consider several preconditioners and iterative methods for solving the linear algebraic system associated with a partial differential equation. Our interest stems from earlier work in Method of Lines (MOL) software for solving kinetics-diffusion equations and a recognition that the solution of the underlying linear system at each timestep is crucial in terms of computational storage and time. We are interested in developing an approach to handle nonsymmetric matrices so that we can deal with convective terms in the partial differential equation (PDE). To examine our methods we consider a model problem which has been used in related work. With regard to the approach there are two aspects: the preconditioner and the iterative method. Among the preconditioners considered are normal form LU factorization and variations related to approximate inverses. The iterative methods include normal form conjugate gradients and related nonsymmetric methods (ORTHOMIN and ORTHODIR). We have found that the use of either an approximate LU factorization or an approximate inverse in combination with normal form conjugate gradient iteration provides an effective approach for solving our model problem. This result suggests potential use of approximate inverses for parallel computation. 5 refs., 4 figs.
Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions
Yaşar, Banu Yılmaz; Özarslan, Mehmet Ali
2016-01-01
In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, K{\\i}ymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin tran...
On Liapunov transformations of linear systems of implicit differential equations
Mazanik, S. A.
1991-01-01
An asymptotical equivalence of linear systems of implicit differential equations is investigated. Sufficient conditions to transform a linear system of implicit differential equations to a system of specal type are given.
Bounded solutions for fuzzy differential and integral equations
Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es
2006-03-01
We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.
Numerical analysis of systems of ordinary and stochastic differential equations
Artemiev, S S
1997-01-01
This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).
Differential equations, associators, and recurrences for amplitudes
Puhlfürst, Georg; Stieberger, Stephan
2016-01-01
We provide new methods to straightforwardly obtain compact and analytic expressions for ɛ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ɛ-orders of a power series solution in ɛ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ɛ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ɛ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we set up our methods to systematically get compact and explicit α‧-expansions of tree-level superstring amplitudes to any order in α‧.
Singular solutions of a singular differential equation
Naito Manabu
2000-01-01
Full Text Available An attempt is made to study the problem of existence of singular solutions to singular differential equations of the type which have never been touched in the literature. Here and are positive constants and is a positive continuous function on . A solution with initial conditions given at is called singular if it ceases to exist at some finite point . Remarkably enough, it is observed that the equation may admit, in addition to a usual blowing-up singular solution, a completely new type of singular solution with the property that Such a solution is named a black hole solution in view of its specific behavior at . It is shown in particular that there does exist a situation in which all solutions of are black hole solutions.
Differential Equations, Associators, and Recurrences for Amplitudes
Puhlfuerst, Georg
2015-01-01
We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different epsilon-orders of a power series solution in epsilon of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the epsilon-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also setup up our tools for computing epsilon-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we apply our methods to systematically get compact and explicit alpha'-expansions of tree-level superstring amplitudes to any order in alpha'.
Polynomial chaos functions and stochastic differential equations
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory
Differential equations, associators, and recurrences for amplitudes
Georg Puhlfürst
2016-01-01
Full Text Available We provide new methods to straightforwardly obtain compact and analytic expressions for ϵ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ϵ-orders of a power series solution in ϵ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ϵ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ϵ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system. Finally, we set up our methods to systematically get compact and explicit α′-expansions of tree-level superstring amplitudes to any order in α′.
Lectures on differential equations for Feynman integrals
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE. (topical review)