This text emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green’s functions, and transform methods.
This text is ideal for students in science, engineering, and applied mathematics.
- Step-by-step approach eases students into the material so they can build on their knowledge base. This text progresses from simple exercises to increasingly powerful mathematical techniques for solving more complicated and realistic physical problems.
- Emphasis on examples and problem solving provides students with a thorough and reasoned approach to problem solving, stressing understanding.
- Physical and mathematical derivations are addressed carefully to ensure that students are aware of assumptions being made.
- Clear and lively writing style engages students and clearly explains details and ideas with patience and sustained enthusiasm.
- 1,000 carefully prepared exercises of varying difficulty provide instructors with flexibility in the selection of material and provide students the background necessary to move on to harder exercises. Answers to selected exercises are available in the back of the book.
- Learning aids are well organized and useful for students.
- More than 200 figures illustrate key concepts. These were prepared by the author using MATLAB®.
- Paragraphs are titled throughout the book for convenient reference.
- Important formulas are set apart in tables.
- Inside covers include important tabulated information.
- Content highlights:
- Pattern formation for reaction-diffusion equations and the Turing instability includes interesting applications such as lift and drag past circular cylinder, reflection and refraction of electromagnetic (light) and acoustic (sound) waves, scattering, dispersive waves, wave guides, fiber optics, and pattern formation.
- Well done treatment of numerical methods for PDE includes Finite difference methods, Fourier/von Neumann stability analysis, heat equation, wave equation, Laplace's equation, and Finite element method (Introduction).
- Presentation of derivation of the diffusion of a pollutant provides instructors with the option early in the text, of a more concise derivation of the one dimensional heat equation.
- Discussion on time dependent heat equations shows students how the time dependent heat equation evolves in time to the steady state temperature distribution.
- Similarity solution for ht heat equation provides students with a concise discussion of similarity solution.
- Green's Functions for Wave and Heat Equations chapter provides students with a presentation of elegant derivations of infinite space Green's functions for heat and wave equations.
- Shock waves chapter presents derivation of the shock velocity presented; diffusive conservation laws introduced; presentations improved on the initiation of a shock and the formation of caustics for the characteristic.
- Stability of systems of ordinary differential equation including eigenvalues of the Jacobian matrix and bifurcations to motivate stability of PDE. This provides students with an expanded presentation on system stability.
- Wave envelope equations—e.g. two-dimensional effects and the modulational instability. This provides students with new material and a brief derivation of the partial differential equation corresponding to a long wave instability.
New to this Edition
- Chapter 2 offers an improved, simpler presentation of the linearity principle, showing that the heat equation is a linear equation.
- Chapter 4 contains a straightforward derivation of the vibrating membrane, an improvement over previous editions.
- Additional simpler exercises now appear throughout the text.
- Hints are offered for many of the exercises in which partial differential equations are solved in chapters 2, 4, 5, 7, and 10—the core of a typical first course. These hints often include the separation for the variables of variables themselves, so the problem is more straightforward for students.