A First Course in Partial Differential Equations with Complex Variables and Transform Methods
Text presents the general properties of partial differential equations such as characteristics, domains of independence and maximum principles. Solutions.
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the vibrating string
The onedimensional wave equation
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absolutely integrable analytic function apply approaches zero approximation boundary conditions boundary value problem bounded Cauchy characteristics circle constant contour converges uniformly coordinates cosh d2u d2u define discontinuity domain of dependence du/dt du/dx dxdy eigenfunctions elliptic Evaluate Example EXERCISES fixed formula Fourier coefficients Fourier series Fourier transform function f(x gives Green's function heat equation Hence hint homogeneous infinite initial conditions initial value problem initial-boundary value problem integrand interval inverse Laplace transform Laplace's equation linear mapping maximum principle means multiply nxdx obtain odd function operator orthogonal Parseval's equation partial derivatives Partial Differential Equations partial sum polynomial positive power series Ppdx respect right-hand side satisfies Schwarz's inequality Section separation of variables sequence series converges Show sin2 sine series sinh sN(x solution u(x square integrable string Suppose Taylor series tion twice continuously differentiable uniqueness wave equation