BOUNDARY VALUE PROBLEMS AND PARTIAL DIFFERENTIAL EQUATIONS
CONTENTS CHAPTER 0 ORDINARY DIFFERENTIAL EQUATIONS 0 1 HOMOGENEOUS LINEAR EQUATIONS 0 2 NONHOMOGENEOUS LINEAR EQUATIONS 0 3 BOUNDARY VALUE PROBLEMS 0 4 SINGULAR BOUNDARY VALUE PROBLEMS 0 5 GREEN’S FUNCTIONS CHAPTER REVIEW MISCELLANEOUS EXERCISES CHAPTER 1 FOURIER SERIES AND INTEGRALS 1 1 PERIODIC FUNCTIONS AND FOURIER SERIES 1 2 ARBITRARY PERIOD AND HALF-RANGE EXPANSIONS 1 3 CONVERGENCE OF FOURIER SERIES 1 4 UNIFORM CONVERGENCE 1 5 OPERATIONS ON FOURIER SERIES 1 6 MEAN ERROR AND CONVERGENCE IN MEAN 1 7 PROOF OF CONVERGENCE 1 8 NUMERICAL DETERMINATION OF FOURIER COEF?CIENTS 1 9 FOURIER INTEGRAL 1 10 COMPLEX METHODS 1 11 APPLICATIONS OF FOURIER SERIES AND INTEGRALS 1 12 COMMENTS AND REFERENCES CHAPTER 2 THE HEAT EQUATION 2 1 DERIVATION AND BOUNDARY CONDITIONS 2 2 STEADY-STATE TEMPERATURES 2 3 EXAMPLE: FIXED END TEMPERATURES 2 4 EXAMPLE: INSULATED BAR 2 5 EXAMPLE: DIFFERENT BOUNDARY CONDITIONS 2 6 EXAMPLE: CONVECTION 2 7 STURM–LIOUVILLE PROBLEMS 2 8 EXPANSION IN SERIES OF EIGENFUNCTIONS 2 9 GENERALITIES ON THE HEAT CONDUCTION PROBLEM 2 10 SEMI-IN?NITE ROD 2 11 IN?NITE ROD 2 12 THE ERROR FUNCTION 2 13 COMMENTS AND REFERENCES CHAPTER 3 THE WAVE EQUATION 3 1 THE VIBRATING STRING 3 2 SOLUTION OF THE VIBRATING STRING PROBLEM 3 3 D’ALEMBERT’S SOLUTION 3 4 ONE-DIMENSIONAL WAVE EQUATION: GENERALITIES 3 5 ESTIMATION OF EIGENVALUES 3 6 WAVE EQUATION IN UNBOUNDED REGIONS 3 7 COMMENTS AND REFERENCES CHAPTER 4 THE POTENTIAL EQUATION 4 1 POTENTIAL EQUATION 4 2 POTENTIAL IN A RECTANGLE 4 3 FURTHER EXAMPLES FOR A RECTANGLE 4 4 POTENTIAL IN UNBOUNDED REGIONS 4 5 POTENTIAL IN A DISK 4 6 CLASSI?CATION AND LIMITATIONS 4 7 COMMENTS AND REFERENCES CHAPTER 5 HIGHER DIMENSIONS AND OTHER COORDINATES 5 1 TWO-DIMENSIONAL WAVE EQUATION: DERIVATION 5 2 THREE-DIMENSIONAL HEAT EQUATION 5 3 TWO-DIMENSIONAL HEAT EQUATION: SOLUTION 5 4 PROBLEMS IN POLAR COORDINATES 5 5 BESSEL’S EQUATION 5 6 TEMPERATURE IN A CYLINDER 5 7 VIBRATIONS OF A CIRCULAR MEMBRANE 5 8 SOME APPLICATIONS OF BESSEL FUNCTIONS 5 9 SPHERICAL COORDINATES; LEGENDRE POLYNOMIALS 5 10 SOME APPLICATIONS OF LEGENDRE POLYNOMIALS 5 11 COMMENTS AND REFERENCES CHAPTER 6 LAPLACE TRANSFORM 6 1 DE?NITION AND ELEMENTARY PROPERTIES 6 2 PARTIAL FRACTIONS AND CONVOLUTIONS 6 3 PARTIAL DIFFERENTIAL EQUATIONS 6 4 MORE DIF?CULT EXAMPLES 6 5 COMMENTS AND REFERENCES CHAPTER 7 NUMERICAL METHODS 7 1 BOUNDARY VALUE PROBLEMS 7 2 HEAT PROBLEMS 7 3 WAVE EQUATION 7 4 POTENTIAL EQUATION 7 5 TWO-DIMENSIONAL PROBLEMS 7 6 COMMENTS AND REFERENCES BIBLIOGRAPHY