 EqWorld The World of Mathematical Equations  ## Handbook of Linear Partial Differential Equations for Engineers and ScientistsSecond Edition, Updated, Revised and Extended

Publisher: Chapman and Hall/CRC Press, Boca Raton-London-New York
Year of Publication: January 27, 2016
Number of Pages: 1632

## Preface to the new edition

Handbook of Linear Partial Differential Equations for Engineers and Scientists, a unique reference for scientists and engineers, contains nearly 4,000 linear partial differential equations with solutions as well as analytical, symbolic, and numerical methods for solving linear equations. First-, second-, third-, fourth-, and higher-order linear equations and systems of coupled equations are considered. Equations of parabolic, hyperbolic, elliptic, mixed, and other types are discussed. A number of new linear equations, exact solutions, transformations, and methods are described. Formulas for effective construction of solutions are given. A number of specific examples where the methods described in the book are used are considered. Boundary value problems and eigenvalue problems are described. Symbolic and numerical methods for solving PDEs with Maple, Mathematica, and MATLAB are considered. All in all, the handbook contains much more linear partial differential equations than any other book currently available.

In selecting the material, the authors have given highest priority to the following major topics:

• Equations and problems that arise in various applications (heat and mass transfer theory, wave theory, elasticity, hydrodynamics, aerodynamics, continuum mechanics, acoustics, electrostatics, electrodynamics, electrical engineering, diffraction theory, quantum mechanics, chemical engineering sciences, control theory, etc.).
• Systems of coupled equations that arise in various fields of continuum mechanics and physics.
• Analytical and symbolic methods for solving linear equations of mathematical physics.
• Equations of general form that depend on arbitrary functions and equations that involve many free parameters; exact solutions of such equations are of major importance for testing numerical and approximate analytical methods.

The second edition has been substantially updated, revised, and expanded. More than 1,500 linear equations and systems with solutions, as well some methods and many examples, have been added, which amounts to over 700 pages of new material (including 250 new pages dealing with methods).

New to the second edition:

• Some second-, third-, fourth-, and higher-order linear PDEs with solutions.
• Systems of coupled partial differential equations with solutions.
• First-order linear PDEs with solutions.
• Some analytical methods including decomposition methods and their applications.
• Symbolic and numerical methods for solving linear PDEs with Maple, Mathematica, and MATLAB.
• Some transformations, asymptotic formulas and solutions.
• Many new examples and figures included for illustrative purposes.
• Some long tables, including tables of various integral transforms.

Note that Chapters 1--12 of the book can be used as a database of test problems for numerical, approximate analytical, and symbolic methods for solving linear partial differential equations and systems of coupled equations. To satisfy the needs of a broad audience with diverse mathematical background, the authors have done their best to avoid special terminology whenever possible. Therefore, some of the methods are outlined in a schematic and somewhat simplified manner with necessary references made to books where these methods are considered in more detail. Many sections are written so that they can be read independently from each other. This allows the reader to get to the heart of the matter quickly.

Separate sections of the book can serve as a basis for practical courses and lectures on equations of mathematical physics and linear PDEs.

We would like to express our keen gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. We are very thankful to Inna Shingareva and Carlos Lizarraga-Celaya, who wrote three chapters (22-24) of the book at our request.

The authors hope that the handbook will prove helpful for a wide audience of researchers, university and college teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, chemistry, economics, and engineering sciences.

## Preface to the first edition

Linear partial differential equations arise in various fields of science and numerous applications, e.g., heat and mass transfer theory, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrostatics, electrodynamics, electrical engineering, diffraction theory, quantum mechanics, control theory, chemical engineering sciences, and biomechanics.

This book presents brief statements and exact solutions of more than 2000 linear equations and problems of mathematical physics. Nonstationary and stationary equations with constant and variable coefficients of parabolic, hyperbolic, and elliptic types are considered. A number of new solutions to linear equations and boundary value problems are described. Special attention is paid to equations and problems of general form that depend on arbitrary functions. Formulas for the effective construction of solutions to nonhomogeneous boundary value problems of various types are given. We consider second-order and higher-order equations as well as the corresponding boundary value problems. All in all, the handbook presents more equations and problems of mathematical physics than any other book currently available.

For the reader's convenience, the introduction outlines some definitions and basic equations, problems, and methods of mathematical physics. It also gives useful formulas that enable one to express solutions to stationary and nonstationary boundary value problems of general form in terms of the Green's function.

Two supplements are given at the end of the book. Supplement A lists properties of the most common special functions (the gamma function, Bessel functions, degenerate hypergeometric functions, Mathieu functions, etc.). Supplement B describes the methods of generalized and functional separation of variables for nonlinear partial differential equations. We give specific examples and an overview application of these methods to construct exact solutions for various classes of second-, third-, fourth-, and higher-order equations (in total, about 150 nonlinear equations with solutions are described). Special attention is paid to equations of heat and mass transfer theory, wave theory, and hydrodynamics as well as to mathematical physics equations of general form that involve arbitrary functions.

The equations in all chapters are in ascending order of complexity. Many sections can be read independently, which facilitates working with the material. An extended table of contents will help the reader find the desired equations and boundary value problems. We refer to specific equations using notation like "1.8.5.2", which means "Equation 2 in Subsection 1.8.5".

To extend the range of potential readers with diverse mathematical backgrounds, the author strove to avoid the use of special terminology wherever possible. For this reason, some results are presented schematically, in a simplified manner (without details), which is however quite sufficient in most applications.

Separate sections of the book can serve as a basis for practical courses and lectures on equations of mathematical physics.

The author thanks Alexei Zhurov for useful remarks on the manuscript.

The author hopes that the handbook will be useful for a wide range of scientists, university teachers, engineers, and students in various areas of mathematics, physics, mechanics, control, and engineering sciences.

The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.