Partial Differential Equations of Mathematical Physics
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Partial derivative
A partial differential equation (PDE) relates the quantities of a multivariate function and its various partial derivatives in an equation. An ordinary partial differential, as discussed in the previous chapter, is a subclass of partial differential equations because ordinary partial differential equations deal with functions with one variable. Partial differential equations are significantly more difficult to solve than ordinary partial differentials because a simple PDE can admit a large class of solutions. For example, for this simple PDE,
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Distributed parameter system
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This chapter explains the exp-function method by which one can find the solution (analytical/semi-analytical) of differential equations. This method was first proposed by He and Wu and was successfully applied to obtain the solitary and periodic solutions of nonlinear partial differential equations. Further, this method was used by many researchers for handling various other equations like stochastic equations, system of partial differential equations, nonlinear evaluation equation of high dimension, difference-differential equation, and nonlinear dispersive long-wave equation. This method is illustrated for partial differential equations as ordinary differential equations are straightforward while solving partial differential equations. The chapter considers a nonlinear partial differential equation to understand briefly the exp-function method. It solves two example problems to make the readers understand the exp-function method.
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In this paper we discuss the first order partial differential equations resolved with any derivatives. At first, we transform the first order partial differential equation resolved with respect to a time derivative into a system of linear equations. Secondly, we convert it into a system of the first order linear partial differential equations with constant coefficients and nonlinear algebraic equations. Thirdly, we solve them by the Fourier transform and convert them into the equivalent integral equations. At last, we extend to discuss the first order partial differential equations resolved with respect to time derivatives and the general scenario resolved with any derivatives.
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Separation of variables
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The partial differential equation plays an important role in Physics, Chemistry, Differential Equations and many more applied subjects. Also various methods of solving partial differential equations are found in literature. In this paper, we use double Elzaki Transform to solve some partial Differential equations.
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This chapter contains sections titled: Elliptic, Parabolic and Hyperbolic Partial Differential Equations Elliptic Partial Differential Equations Parabolic Partial Differential Equations Hyperbolic Partial Differential Equations Problems
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Preface to the Second Edition / Preface to the First Edition / First Order Partial Differential Equations / Second Order Partial Differential Equations / Elliptic Differential Equations / Parabolic Differential Equations / Hyperbolic Differential Equations / Integral Transform and Green Function Methods / Integral Equations / Numerical Solutions of Partial Differential Equations / Answers to Key Exercise / Appendix A / Appendix B / Bibliography / Index.
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