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costa p select ideas partial differential equations 2021

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Textbook in PDF format Condensed Summary by Chapter: Review the definitions of the divergence, gradient, and curl in three spatial dimensions.Use Green’s, Stokes’, and Gauss’s Divergence Theorem to help establish Faraday’s Law, Coulomb’s Law, and Ampere’s Law. Use these results to present Maxwell’s equations. This is the starting point for the book’s approach to PDEs. Develop classical (separation of variables) and modern methods (Green’s functions) solution of Laplace’s equation. Introduce the Dirichlet, Neumann, and Robin boundary value variations of Laplace’s equation along with Poisson’s equation. Prove Green’s Representation Theorem for smooth functions on general domains in n . Provide examples of analytic solutions over regular bounded domains in 2 . Introduce the idea of analytic functions (namely those functions which have convergent Taylor series) and their rôle in solving PDEs. Detail Fourier series, its application in representing periodic functions that need not be continuous, and its use in the solution of Laplace’s equation over varying geometry. Begin the discussion of how functional analysis, through function spaces, help solve PDEs. Show how Green’s functions solve Poisson’s equation. Introduce the heat equation and the use of Bessel functions in its solution over circular geometries. Introduce the Fourier transform and show how it is used to solve the heat equation in n. Use the Fourier transform to derive d’Alembert’s solution for the homogeneous and inhomogeneous wave equations in . Construct the various properties of the Fourier transform and establish a key set of Fourier transform pairs. Solve the string equation (i.e., the one–dimensional wave equation over a finite interval), the circular vibrating drum equation, and the vibrating volume equation over finite domains. Use the Fourier transform to solve the wave equation over 3 . Detail Huygen’s and Duhamel’s principles to arrive at spherical means solutions to the three–dimensional inhomogeneous wave equation. Employ the method of descent to arrive at the solution of the two–dimensional wave equation. Develop solutions of Maxwell’s equations via the three–dimensional wave solutions. Introduce examples of nonlinear PDEs which are solved via “transformations.” The first example, the nonlinear Klein–Gordon (nKG) equation, can be solved by the classical transformation method separation of variables. Burgers’ equation is reduced to the heat equation via a linearizing change of variables: The Hopf–Cole Transformation. The well-studied Korteweg de Vries (KdV) equation is solved using the inverse scattering transform which reduces solutions of the KdV to solutions of the linear Airy’s equation. Numerical methods. The system of nonlinear PDEs which model stimulated Raman scattering are established. Functional analysis is utilized to show that a particular finite difference scheme is both stable and convergent. Moreover, a priori conditions are placed on the temporal step size to insure stability of the finite difference method. It is demonstrated that the system is a conserved Hamiltonian. A series of computations contrasting the effects of varying initial conditions are illustrated. Discussion of the physical implications of the calculations is presented. The examples in Chapter 6 illustrate that a variety of nonlinear PDEs can be solved by a change of variables which transform the original nonlinear equation into a linear PDE

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costa p select ideas partial differential equations 2021

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