ESSENTIALS OF APPLIED MATHEMATICS
CONTENTS 1 PARTIAL DIFFERENTIAL EQUATIONS IN ENGINEERING INTRODUCTORY COMMENTS FUNDAMENTAL CONCEPTS THE HEAT CONDUCTION (OR DIFFUSION) EQUATION RECTANGULAR CARTESIAN COORDINATES CYLINDRICAL COORDINATES SPHERICAL COORDINATES THE LAPLACIAN OPERATOR THE VIBRATING STRING VIBRATING MEMBRANE LONGITUDINAL DISPLACEMENTS OF AN ELASTIC BAR 2 THE FOURIER METHOD: SEPARATION OF VARIABLES HEAT CONDUCTION SEPARATION OF VARIABLES SUPERPOSITION ORTHOGONALITY LESSONS PROBLEMS 2 1 10 ORTHOGONALITY 2 1 11 LESSONS 2 1 13 GETTING TO ONE NONHOMOGENEOUS CONDITION 2 1 14 SEPARATION OF VARIABLES 2 1 16 SUPERPOSITION 2 1 17 ORTHOGONALITY 2 1 18 LESSONS 2 1 20 RELOCATING THE NONHOMOGENEITY 2 1 21 SEPARATING VARIABLES 2 1 22 SUPERPOSITION 2 1 23 ORTHOGONALITY 2 1 24 LESSONS VIBRATIONS FURTHER READING 3 ORTHOGONAL SETS OF FUNCTIONS ORTHOGONALITY OF VECTORS FUNCTIONS ORTHONORMAL SETS OF FUNCTIONS AND FOURIER SERIES STURM–LIOUVILLE PROBLEMS: ORTHOGONAL FUNCTIONS ORTHOGONALITY OF EIGENFUNCTIONS 4 SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS GENERAL SERIES SOLUTIONS DE?NITIONS ORDINARY POINTS AND SERIES SOLUTIONS LESSONS: FINDING SERIES SOLUTION FOR DIFFERENTIAL EQUATIONS WITH REGULAR SINGULAR POINTS LOGARITHMS AND SECOND SOLUTIONS BESSEL FUNCTIONS HERE ARE THE RULES LEGENDRE FUNCTIONS ASSOCIATED LEGENDRE FUNCTIONS 5 SOLUTIONS USING FOURIER SERIES AND INTEGRALS CONDUCTION (OR DIFFUSION) PROBLEMS VIBRATIONS PROBLEMS FOURIER INTEGRALS PROBLEM 6 INTEGRAL TRANSFORMS: THE LAPLACE TRANSFORM THE LAPLACE TRANSFORM SOME IMPORTANT TRANSFORMS EXPONENTIALS SHIFTING IN THE S -DOMAIN SHIFTING IN THE TIME DOMAIN SINE AND COSINE HYPERBOLIC FUNCTIONS POWERS OF T: TM HEAVISIDE STEP THE DIRAC DELTA FUNCTION TRANSFORMS OF DERIVATIVES 6 2 10 LAPLACE TRANSFORMS OF INTEGRALS 6 2 11 DERIVATIVES OF TRANSFORMS LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEF?CIENTS SOME IMPORTANT THEOREMS FINAL VALUE THEOREM CONVOLUTION PARTIAL FRACTIONS NONREPEATING ROOTS QUADRATIC FACTORS: COMPLEX ROOTS 7 COMPLEX VARIABLES AND THE LAPLACE INVERSION INTEGRAL BASIC PROPERTIES INTEGRALS DIFFUSION OR CONDUCTION PROBLEMS A PRELIMINARY EXAMPLE: FOURIER SINE TRANSFORM GENERALIZATION: THE STURM–LIOUVILLE TRANSFORM: THEORY THE INVERSE TRANSFORM 10 1 1 REGULAR PERTURBATION 10 1 2 SINGULAR PERTURBATION