Introduction To Partial Differential Equations By Sankara Rao 40
Introduction to Partial Differential Equations by Sankara Rao 40
Partial differential equations (PDEs) are equations that involve partial derivatives of unknown functions of two or more variables. They are widely used to model various physical phenomena, such as heat conduction, fluid flow, wave propagation, and electromagnetism. PDEs are also important in mathematics, as they often arise from the study of geometry, analysis, and differential equations.
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One of the books that provides a comprehensive introduction to PDEs is Introduction to Partial Differential Equations by K. Sankara Rao. This book covers the fundamental concepts, the underlying principles, and various well-known mathematical techniques and methods to solve PDEs with different initial and boundary conditions. The book is divided into six chapters, as follows:
Chapter 1: This chapter introduces the basic definitions and classifications of PDEs, such as linear and nonlinear, homogeneous and nonhomogeneous, elliptic, parabolic, and hyperbolic. It also discusses some examples of PDEs from physics and engineering, such as the Laplace equation, the heat equation, and the wave equation.
Chapter 2: This chapter deals with the method of separation of variables, which is one of the most common methods to solve PDEs with separable boundary conditions. The chapter explains how to separate the variables in Cartesian, cylindrical, and spherical coordinates, and how to use Fourier series and eigenvalue problems to find the solutions.
Chapter 3: This chapter introduces the concept of Laplace transform and its properties, and shows how to use it to solve linear PDEs with constant coefficients and initial conditions. The chapter also covers some applications of Laplace transform to solve PDEs involving discontinuous or periodic functions.
Chapter 4: This chapter presents another powerful tool to solve linear PDEs with constant coefficients and boundary conditions, which is the Fourier transform. The chapter explains the definition and properties of Fourier transform, and how to use it to solve PDEs in one or two dimensions. The chapter also discusses some applications of Fourier transform to solve PDEs involving infinite or semi-infinite domains.
Chapter 5: This chapter focuses on the method of characteristics, which is a geometric method to solve first-order PDEs or quasi-linear PDEs. The chapter illustrates how to find the characteristic curves or surfaces along which the solution is constant or satisfies an ordinary differential equation. The chapter also covers some examples of PDEs that can be solved by this method, such as the transport equation, the Burgers equation, and the eikonal equation.
Chapter 6: This chapter introduces the concept of Green's function and its applications to solve linear PDEs with nonhomogeneous boundary conditions. The chapter explains how to construct Green's function for various types of PDEs and domains, such as the Poisson equation, the Helmholtz equation, and the Dirichlet problem. The chapter also discusses some properties and applications of Green's function, such as the superposition principle, the uniqueness theorem, and the representation formula.
The book is supported by miscellaneous examples and exercises to help students understand and apply the concepts and techniques for solving PDEs. The book also provides some references for further reading at the end of each chapter. The book is suitable for undergraduate and postgraduate students of mathematics, physics, engineering, and other related disciplines who want to learn about PDEs.
The book is available in paperback format from Prentice-Hall Of India Pvt. Limited, or in PDF format from various online sources. The book has 508 pages in its latest edition (2010), which is also known as Sankara Rao 40.
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