Fourier Analysis and Boundary Value ProblemsFourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics. A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.
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From inside the book
Results 1-5 of 77
... implies that XT = u = kuxx = k X"T for (x, t) in D. If we further assume that X(x) # 0 for 0 < x < a. and that T(t) # 0 for t > 0 we obtain T/ k X" T - X" Now, the only way for a function of t to equal a function of x is for both of ...
... imply that A + B = 0 Aev-Fa + Be-v-Ha = 0 which in turn imply that A = B = 0. This leads to the trivial solution u = TX = 0, which is indeed a solution but only when f = 0. If X = 0 the original equation reduces to X" = 0, so that X(x) ...
... implies that f can be extended as a continuous function to each closed interval [xi_1, xi). Thus, f is bounded, and, since it has only a finite number of discontinuities, it is integrable by Lebesgue's theorem—in this simple particular ...
... in the graph of DN (t – X) is approximately 6(2N + 1) = 27t. Then (8) implies that x+8 f 1 (x) .x--ó sN (x) * 2nt f(t) DN(t – x) dt = |. DN (t — X) dt = f(x) 7t. JY-8 27 J-5 2N + 1 f(t) | , a ~ A A 38 FOURIER SERIES CHAP. 2.
... imply its existence in the general case. But in 1906 Maxime Bôcher (1867–1918), of Harvard University—who was not familiar with the work of either Wilbraham or Gibbs—showed that the partial sum of the Fourier series of an arbitrary ...
Contents
1 | |
23 | |
84 | |
CHAPTER 4 GENERALIZED FOURIER SERIES | 133 |
CHAPTER 5 THE WAVE EQUATION | 165 |
CHAPTER 6 ORTHOGONAL SYSTEMS | 207 |
CHAPTER 7 FOURIER TRANSFORMS | 237 |
CHAPTER 8 LAPLACE TRANSFORMS | 266 |
CHAPTER 10 BOUNDARY VALUE PROBLEMS WITH CIRCULAR SYMMETRY | 351 |
CHAPTER 11 BOUNDARY VALUE PROBLEMS WITH SPHERICAL SYMMETRY | 410 |
CHAPTER 12 DISTRIBUTIONS AND GREENS FUNCTIONS | 451 |
APPENDIX A UNIFORM CONVERGENCE | 506 |
APPENDIX B IMPROPER INTEGRALS | 518 |
APPENDIX C TABLES OF FOURIER AND LAPLACE TRANSFORMS | 535 |
APPENDIX D HISTORICAL BIBLIOGRAPHY | 539 |
Index | 543 |