Integral Equations (& Boundary Value Problems)

CONTENTS- INTEGRAL EQUATIONS,
Chapter-1: Basic Concepts
1.1 Integral Equation
1.2 Differentiation of a Function Under an Integral Sign
1.3 Relation Between Differential and Integral Equations
Chapter-2: Solution of Integral Equations
2.1 Solution of Nonhomogeneous Volterra's Integral Equation of Second kind by the Method of Successive Substitution
2.5 Solution of the Fredholm Integral Equation by the Method of Successive Substitutions
2.6 Iterated Kernels
2.7 Solution of the Fredholm Integral Equation by the Method of Successive Approximation
2.8 Reciprocal Functions
2.9 Volterra's Solution of Fredholm's Equation
Chapter-3: Fredholm Integral Equations
3.1 Fredholm First Theorem
3.2 Prove that the solution
3.3 Every Zero of Fredholm Function D(l) is a Pole of the Resolvent Kernel
3.4 If a Real Kernel K (x, ) has a Complex Eigen Value iv, then it Also Contains the Conjugate Eigen Value to 0– iv
3.5 Hadamard's Lemma
3.6 Convergence Proof
3.7 Fredholm Second Theorem
3.8 Fredholm's Associated Equation
3.9 Characteristic Solutions
3.10 Fredholm's Third Theorem
3.11 Solution of the Homogeneous Integral Equation
3.12 If D(0) 0 and D (x, ; 0) /0, then for a Proper Choice of 0,(x) D (x, 0; 0) is a Continuous Solution of the Homogeneous Integral Equation
3.13 Fundamental Functions
3.14 Integral Equations with Degenerate Kernels
Chapter-4: Hilbert Schmidt Theory
4.1 All Iterated Kernels of a Symmetric Kernel are also Symmetric
4.2 Orthogonality
4.3 Orthogonality of Fundamental Functions
4.4 Eigen Values of Symmetric Kernel are Real
4.10 Fourier Series of Power of the Eigen Values of the Iterated Kernel
4.11 Hilbert-Schmidt Theorem
4.12 The inequalities of Schwarz and Minkowski
4.13 Hilbert's Theorem
4.14 Complete Normalized Orthogonal System of Characteristic Functions
4.15 Coefficients of the Continuous Function f (x)
4.16 Complete Normalized Orthogonal System of Fundamental Functions
4.17 Bessel Inequality
4.18 Riesz-Fischer Theorem
4.19 Representation by a linear Combination of the Characteristic Functions
4.20 Schmidt's Solution of the Non-Homogeneous Integral Equation
4.21 Solution of the Fredholm Integral Equation of first Kind
Chapter-5: Application of Integral Equations
5.1 Introduction
5.2 Initial Value Problem
5.3 Boundary Value Problems
5.4 Deformation of a Rod
5.5 Determination of Periodic Solutions
5.6 Green's Function
5.7 Construction of Green's Function
5.8 Particular Case
5.9 Influence Function
5.10 Construction of Green's Function when the Boundary Value Problem Contains a Parameter
5.11 Longitudinal Vibrations of a Rod
Chapter-6: Singular Integral Equations
6.1 Introduction
6.2 Abel Integral Equation
6.3 Particular Case
6.4 Weakly Singular Kernel
6.5 Iteration of the Singular Equation
6.6 Fredholm Operator
6.7 Equivalence of the Fredholm Integral Equation and the Iterated Equation
6.8 Prove that the Eigenvalues 0 and p of the Kernels k and kp are of the Same Rank
6.9 If a Number is an Eigenvalue of the Iterated Kernel kp (x, ), then atleast one of the Distinct Numbers
6.10 Integral Equation in an Infinite Interval
6.11 Cauchy Principal Integral
6.12 Cauchy Type Integral
6.13 Cauchy Integral on the Path of Integration
6.14 Plemelj Formulae
6.15 The Plemelj –Privalov Theorem
6.16 Poincare'-Bertrand Transformation Formula for Iterated Singular Integrals
6.17 Application of the Calculus of Residues
6.18 Hilbert Kernel
6.19 Solution of the Cauchy-type Singular Integral Equation
Chapter-7: Integral