AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB

By B. LUVAN

CONTENTS PRELUDE TO CHAPTER INTRODUCTION WHAT ARE PARTIAL DI?ERENTIAL EQUATIONS? PDES WE CAN ALREADY SOLVE INITIAL AND BOUNDARY CONDITIONS LINEAR PDES—DE?NITIONS LINEAR PDES—THE PRINCIPLE OF SUPERPOSITION SEPARATION OF VARIABLES FOR LINEAR, HOMOGENEOUS PDES EIGENVALUE PROBLEMS PRELUDE TO CHAPTER THE BIG THREE PDES SECOND-ORDER, LINEAR, HOMOGENEOUS PDES WITH CONSTANT CO- E?CIENTS THE HEAT EQUATION AND DI?USION THE WAVE EQUATION AND THE VIBRATING STRING INITIAL AND BOUNDARY CONDITIONS FOR THE HEAT AND WAVE EQUA- TIONS LAPLACE’S EQUATION—THE POTENTIAL EQUATION USING SEPARATION OF VARIABLES TO SOLVE THE BIG THREE PDES FOURIER SERIES PROPERTIES OF SINE AND COSINE THE FOURIER SERIES THE FOURIER SERIES, CONTINUED THE FOURIER SERIES—PROOF OF POINTWISE CONVERGENCE FOURIER SINE AND COSINE SERIES COMPLETENESS SOLVING THE BIG THREE PDES ON FINITE DOMAINS SOLVING THE HOMOGENEOUS HEAT EQUATION FOR A FINITE ROD SOLVING THE HOMOGENEOUS WAVE EQUATION FOR A FINITE STRING SOLVING THE HOMOGENEOUS LAPLACE’S EQUATION ON A RECTANGULAR DOMAIN NONHOMOGENEOUS PROBLEMS CHARACTERISTICS FIRST-ORDER PDES WITH CONSTANT COE?CIENTS FIRST-ORDER PDES WITH VARIABLE COE?CIENTS THE IN?NITE STRING CHARACTERISTICS FOR SEMI-IN?NITE AND FINITE STRING PROBLEMS GENERAL SECOND-ORDER LINEAR PDES AND CHARACTERISTICS INTEGRAL TRANSFORMS THE LAPLACE TRANSFORM FOR PDES FOURIER SINE AND COSINE TRANSFORMS THE FOURIER TRANSFORM THE IN?NITE AND SEMI-IN?NITE HEAT EQUATIONS DISTRIBUTIONS, THE DIRAC DELTA FUNCTION AND GENERALIZED FOURIER TRANSFORMS PROOF OF THE FOURIER INTEGRAL FORMULA SPECIAL FUNCTIONS AND ORTHOGONAL POLYNOMIALS THE SPECIAL FUNCTIONS AND THEIR DI?ERENTIAL EQUATIONS ORDINARY POINTS AND POWER SERIES SOLUTIONS; CHEBYSHEV, HER- MITE AND LEGENDRE POLYNOMIALS THE METHOD OF FROBENIUS; LAGUERRE POLYNOMIALS INTERLUDE: THE GAMMA FUNCTION BESSEL FUNCTIONS RECAP: A LIST OF PROPERTIES OF BESSEL FUNCTIONS AND ORTHOGONAL POLYNOMIALS STURM–LIOUVILLE THEORY AND GENERALIZED FOURIER SERIES STURM–LIOUVILLE PROBLEMS REGULAR AND PERIODIC STURM–LIOUVILLE PROBLEMS SINGULAR STURM–LIOUVILLE PROBLEMS; SELF-ADJOINT PROBLEMS THE MEAN-SQUARE OR L2 NORM AND CONVERGENCE IN THE MEAN GENERALIZED FOURIER SERIES; PARSEVAL’S EQUALITY AND COMPLETE- NESS PDES IN HIGHER DIMENSIONS PDES IN HIGHER DIMENSIONS: EXAMPLES AND DERIVATIONS THE HEAT AND WAVE EQUATIONS ON A RECTANGLE; MULTIPLE FOURIER SERIES LAPLACE’S EQUATION IN POLAR COORDINATES: POISSON’S INTEGRAL FORMULA THE WAVE AND HEAT EQUATIONS IN POLAR COORDINATES PROBLEMS IN SPHERICAL COORDINATES THE IN?NITE WAVE EQUATION AND MULTIPLE FOURIER TRANSFORMS POSTLUDE: EIGENVALUES AND EIGENFUNCTIONS OF THE LAPLACE OPER- ATOR; GREEN’S IDENTITIES FOR THE LAPLACIAN 10 NONHOMOGENEOUS PROBLEMS AND GREEN’S FUNCTIONS 10 1 GREEN’S FUNCTIONS FOR ODES 10 2 GREEN’S FUNCTION AND THE DIRAC DELTA FUNCTION 10 3 GREEN’S FUNCTIONS FOR ELLIPTIC PDES (I): POISSON’S EQUATION IN TWO DIMENSIONS 10 4 GREEN’S FUNCTIONS FOR ELLIPTIC PDES (II): POISSON’S EQUATION IN THREE DIMENSIONS; THE HELMHOLTZ EQUATION 10 5 GREEN’S FUNCTIONS FOR EQUATIONS OF EVOLUTION 11 NUMERICAL METHODS 11 1 FINITE DI?ERENCE APPROXIMATIONS FOR ODES 11 2 FINITE DI?ERENCE APPROXIMATIONS FOR PDES 11 3 SPECTRAL METHODS AND THE FINITE ELEMENT METHOD