This book provides ideas for implementing Wolfram Mathematica to solve linear integral equations. The book introduces necessary theoretical information about exact and numerical methods of solving integral equations. Every method is supplied with a large number of detailed solutions in Wolfram Mathematica. In addition, the book includes tasks for individual study.
This book is a supplement for students studying “Integral Equations”. In addition, the structure of the book with individual assignments allows to use it as a base for various courses.
Author(s): Vladimir Ryzhov, Tatiana Fedorova, Kirill Safronov, Shaharin Anwar Sulaiman, Samsul Ariffin Abdul Karim
Publisher: Springer
Year: 2022
Language: English
Pages: 200
City: Singapore
Introduction
References
Contents
1 Fundamentals. Classification of Integral Equations
1.1 Basic Types of Integral Equations: A Solution of Integral Equation
1.1.1 Fredholm Equation of the Second Kind
1.1.2 Fredholm Equation of the First Kind
1.1.3 Volterra Equation of the Second Kind
1.1.4 Volterra Equation of the First Kind
1.2 Equations with a Weak Singularity
1.3 Abel Problem: Abel Integral Equation
1.4 Solution of Integral Equations by the Differentiation Method
References
2 Integral Equations with Difference Kernels
2.1 Difference Kernel Concept. Solution of Integral Equations with Difference Kernels by the Method of Differentiation
2.2 Solution of Integral Equations and Systems of Volterra Integral Equations with Difference Kernels Using the Laplace Transform
2.2.1 Solving Volterra Integral Equations with Difference Kernels Using the Laplace Transform
2.2.2 Solving Systems of Volterra Integral Equations with Difference Kernels Using the Laplace Transform
2.2.3 Solving Integro-Differential Equations with Difference Kernels Using the Laplace Transform
2.3 Solving Fredholm Integral Equations with Difference Kernels Using the Fourier Transform
References
3 Fredholm Theory
3.1 Solution of Fredholm Integral Equations by the Resolvent Method: Method of Fredholm Determinants
3.2 Iterated Kernels Method
3.3 Characteristic Numbers and Eigenfunctions. Solution of Homogeneous Fredholm Integral Equations with Degenerate Kernel
3.4 Solution of Fredholm Inhomogeneous Integral Equations with a Degenerate Kernel. Fredholm’s Theorems
References
4 Symmetric Integral Equations
4.1 Construction of an Orthonormal System of Eigenfunctions of a Symmetric Kernel
4.2 Representation of the Solution as Expansion in Terms of Orthonormal Eigenfunctions of a Symmetric Kernel
References
5 Approximate Methods for Solving Integral Equations
5.1 Approximate Solution of the Fredholm Equation by Replacing the Integral by a Finite Sum
5.2 Successive Approximation Method
5.3 Bubnov–Galerkin Method
5.3.1 Method of Replacing a Kernel with a Degenerate One
References
6 Individual Tasks. Passing the Final Test After Completing the Course
References
