Applied Integral Equations

"There is a vital role of differential and integral equations in studying different types of real-world problems to study the behavior of the issues. Thus, it becomes essential to know the various methods of finding solutions of the integral equation in explicit form. For the integral equations whose solutions cannot be found in explicit form, one has to study the properties of solutions of the given differential equation to guess an approximate solution.

This textbook entitled "Applied Integral Equations" is intended to study the methods of finding the explicit solutions of integral equations where ever possible and in the absence of finding an exact solution. It is intended to study the properties of solutions of the given integral equations. This book contains 08 chapters.

Chapter-1 discusses the introduction to integral equations, classification of integral equations, Relation between linear differential equations and Volterra integral equation, Nonlinear equation and solution of an integral equation. Chapter-2 discusses the existence and uniqueness theorems of Integral equations, Successive approximation, Iterated Functions, Reciprocal functions, Volterra Solution of Fredholm's equation, Discontinuous Solution, Fredholm equations with separable kernels and Resolvent Kernel. Chapter-3 discusses the Fredholm equation as a limit of a finite system of linear equations, Hadamard's Theorem, Fredholm's two fundamental relations, Fredholm's solution of the Integral equation for different , Characteristic numbers and basic functions, the associated Homogenous integral equations, the orthogonality theorem, Kernels of the form , Eigen Values and eigenfunctions, Fredholm integral equation of the second kind, Eigenvalues for non-separable kernels, Volterra Integral Equation, Solution by the Resolvent kernel and Method of successive approximation. Chapter-4 discusses the Applications of Fredholm theory, Free vibration of an elastic string, The differential equation of the problem, Reduction to a dimensional BVP, Solution of the boundary value problem, Construction of Green function, Equivalence between the Boundary value problem and Linear integral equations, Constrained vibrations of an elastic String, Equivalence between boundary value problem and Linear integral equations and Remark on the solution of the BVP. Chapter-5 discusses the Hilbert-Schmidt Theory that includes Iterations of symmetric kernels, Orthogonality theorem, An existence theorem for the nonlinear integral equation of Fredholm type and the equation of Bratu. Chapter-6 discusses the Fredholm alternatives, An example of Picard's method, Powers of an integral operator, Iterated kernels, Neumann series, A remark on the convergence of the iterative method, Differentiation of function under an integral sign, Relation between differential and integral equation, The Fredholm alternatives and the Fredholm alternative theorem. Chapter-7 discusses the method of undetermined coefficients that includes approximation methods of undetermined coefficients, the method of collocation, the method of weighting functions, the method of least squares and approximation of the kernel.

This book is based on syllabi of the theory of integral equations prescribed for the undergraduate and postgraduate students of mathematics and PhD students in different institutions and universities of India and abroad. This book will be helpful for the competitive examinations as well.