DIFFERENTIAL AND INTEGRAL EQUATIONS
CONTENTS SOME PRELIMINARIES INTEGRAL EQUATIONS AND PICARD’S METHOD INTEGRAL EQUATIONS AND THEIR RELATIONSHIP TO DI?ERENTIAL EQUATIONS PICARD’S METHOD EXISTENCE AND UNIQUENESS FIRST-ORDER DI?ERENTIAL EQUATIONS IN A SINGLE INDEPENDENT VARIABLE TWO SIMULTANEOUS EQUATIONS IN A SINGLE VARIABLE A SECOND-ORDER EQUATION THE HOMOGENEOUS LINEAR EQUATION AND WRONSKIANS SOME LINEAR ALGEBRA WRONSKIANS AND THE LINEAR INDEPENDENCE OF SOLUTIONS OF THE SECOND-ORDER HOMOGENEOUS LINEAR EQUATION THE NON-HOMOGENEOUS LINEAR EQUATION THE METHOD OF VARIATION OF PARAMETERS GREEN’S FUNCTIONS FIRST-ORDER PARTIAL DI?ERENTIAL EQUATIONS CHARACTERISTICS AND SOME GEOMETRICAL CONSIDERATIONS SOLVING CHARACTERISTIC EQUATIONS GENERAL SOLUTIONS FITTING BOUNDARY CONDITIONS TO GENERAL SOLUTIONS PARAMETRIC SOLUTIONS AND DOMAINS OF DE?NITION A GEOMETRIC INTERPRETATION OF AN ANALYTIC CONDITION SECOND-ORDER PARTIAL DI?ERENTIAL EQUATIONS CHARACTERISTICS REDUCTION TO CANONICAL FORM PROBLEMS INVOLVING BOUNDARY CONDITIONS APPENDIX: TECHNIQUE IN THE USE OF THE CHAIN RULE THE DI?USION AND WAVE EQUATIONS AND THE EQUATION OF LAPLACE THE EQUATIONS TO BE CONSIDERED ONE-DIMENSIONAL HEAT CONDUCTION TRANSVERSE WAVES IN A ?NITE STRING SEPARATED SOLUTIONS OF LAPLACE’S EQUATION IN POLAR CO-ORDINATES AND LEGENDRE’S EQUATION THE DIRICHLET PROBLEM AND ITS SOLUTION FOR THE DISC RADIALLY SYMMETRIC SOLUTIONS OF THE TWO-DIMENSIONAL WAVE EQUATION AND BESSEL’S EQUATION EXISTENCE AND UNIQUENESS OF SOLUTIONS, WELL-POSED PROBLEMS APPENDIX: PROOF OF THE MEAN VALUE THEOREM FOR HARMONIC FUNCTIONS THE FREDHOLM ALTERNATIVE A SIMPLE CASE SOME ALGEBRAIC PRELIMINARIES THE FREDHOLM ALTERNATIVE THEOREM A WORKED EXAMPLE HILBERT–SCHMIDT THEORY EIGENVALUES ARE REAL AND EIGENFUNCTIONS CORRESPONDING TO DISTINCT EIGENVALUES ARE ORTHOGONAL ORTHONORMAL FAMILIES OF FUNCTIONS AND BESSEL’S INEQUALITY SOME RESULTS ABOUT EIGENVALUES DEDUCIBLE FROM BESSEL’S INEQUALITY DESCRIPTION OF THE SETS OF ALL EIGENVALUES AND ALL EIGENFUNCTIONS THE EXPANSION THEOREM 10 ITERATIVE METHODS AND NEUMANN SERIES AN EXAMPLE OF PICARD’S METHOD POWERS OF AN INTEGRAL OPERATOR ITERATED KERNELS NEUMANN SERIES A REMARK ON THE CONVERGENCE OF ITERATIVE METHODS 11 THE CALCULUS OF VARIATIONS THE FUNDAMENTAL PROBLEM SOME CLASSICAL EXAMPLES FROM MECHANICS AND GEOMETRY THE DERIVATION OF EULER’S EQUATION FOR THE FUNDAMENTAL PROBLEM THE SPECIAL CASE F = F(Y, Y?) WHEN F CONTAINS HIGHER DERIVATIVES OF Y WHEN F CONTAINS MORE DEPENDENT FUNCTIONS WHEN F CONTAINS MORE INDEPENDENT VARIABLES INTEGRAL CONSTRAINTS NON-INTEGRAL CONSTRAINTS 11 10 VARYING BOUNDARY CONDITIONS 12 THE STURM–LIOUVILLE EQUATION SOME ELEMENTARY RESULTS ON EIGENFUNCTIONS AND EIGENVALUES THE STURM–LIOUVILLE THEOREM DERIVATION FROM A VARIATIONAL PRINCIPLE SOME SINGULAR EQUATIONS THE RAYLEIGH–RITZ METHOD 13 SERIES SOLUTIONS POWER SERIES AND ANALYTIC FUNCTIONS ORDINARY AND REGULAR SINGULAR POINTS POWER SERIES SOLUTIONS NEAR AN ORDINARY POINT EXTENDED POWER SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT: THEORY EXTENDED POWER SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT: PRACTICE THE METHOD OF FROBENIUS SUMMARY APPENDIX: THE USE OF COMPLEX VARIABLES 14 TRANSFORM METHODS THE FOURIER TRANSFORM APPLICATIONS OF THE FOURIER TRANSFORM THE LAPLACE TRANSFORM APPLICATIONS OF THE LAPLACE TRANSFORM APPLICATIONS INVOLVING COMPLEX ANALYSIS APPENDIX: SIMILARITY SOLUTIONS 15 PHASE-PLANE ANALYSIS THE PHASE-PLANE AND STABILITY LINEAR THEORY SOME NON-LINEAR SYSTEMS LINEARISATION