Heat Conduction
Published on 31. March 1991
Book
Hardback
330 pages
978-0-632-01716-4 (ISBN)
Description
One of the most important partial differential equations in applied mathematics is the heat or diffusion equation. Its importance in the modeling of heat conduction, diffusional processes, and flow through a porous medium is well known. It arises from a probabilistic framework and emerges as the simplest approximation to bulk processes governed at the microscopic level by random spatial variations. Heat Conduction provides a balanced account of solutions and results for the heat equation and serves as a modern undergraduate text that reflects the importance of the heat equation in applied mathematics and mathematical modeling. The first two chapters of the book are introductory and summarize the essential elements of heat flow, diffusion, the mathematical formulation, and simple general results. The next two chapters develop exact analytical solutions, obtained by Laplace transforms and Fourier series, for infinite and finite media problems respectively. Other chapters deal with approximate analytical solutions based on the heat-balance integral method, numerical methods for the heat equation, and simple heat conduction moving boundary problems.
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Content
Preface. List of Symbols. INTRODUCTION. Historical Introduction. Physical Derivation of the One Dimensional Heat Equation. One Dimensional Source Solution. Probabilistic Derivation of the One Dimensional Heat Equation. Initial and Boundary Conditions for One Dimensional Problems. Three Dimensional Heat Conduction. Problems. MATHEMATICAL PRELIMINARIES. Introduction. The Existence of Solutions of the Heat Equation. Maximum Principle and Uniqueness Theorem. Laplace Transforms. Fourier Series. Green's Functions. Problems. EXACT ANALYTICAL SOLUTIONS FOR SEMI-INFINITE MEDIA. Introduction. Surface x = 0 with Prescribed Temperature g(t). Surface x = 0 with Prescribed Flux h(t). Newton Cooling at Surface x = 0 Into a Medium of Prescribed Temperature T0(t). Wedge Initially at Zero Temperature and Both Faces at the Constant Temperature v0. Special Wedge Angles q0 = p/2, p and 2p. Problems. EXACT ANALYTICAL SOLUTIONS BY FOURIER SERIES. Introduction. Linear Flow in (0,l) with Prescribed Surface Temperatures. Linear Flow in (0,l) with Prescribed Surface Fluxes. Linear Flow in (0,l) with Newton Heat Loss Into Media at Prescribed Temperatures. Radial Flow in Solid Circular Cylinders and Spheres. Flow in a Rectangle with Prescribed Boundary Temperature. Problems. APPROXIMATE ANALYTICAL SOLUTIONS BY HEAT-BALANCE. Introduction. Semi-Infinite Media with x = 0 at Constant Temperature v0. Semi-Infinite Media with Constant Flux w0 Along x = 0. Semi-Infinite Media with Newton Cooling at x = 0 into a Medium at Constant Temperature T0. One Dimensional Problems for Finite Media. Radial Flow in Solid Circular Cylinders and Spheres. Problems. NUMERICAL SOLUTIONS. Introduction. Explicit Finite Difference Methods. Implicit Finite Difference Methods. Finite Element Method. Boundary Integral Method. Sample Programs. Problems. Melting or Freezing Moving Boundary Problems. Introduction. Exact Solution for Prescribed Surface Temperature. Pseudo Steady State and Large a Approximations. Integral Formulation, Bounds and Integral Iteration. Approximate Analytical Solutions by Heat-Balance. Enthalpy Formulation and Numerical Solution. Problems. Table of Laplace Transforms. Bibliography. Index.