An Elementary Course in Partial Differential Equations
An Elementary Course in Partial Differential Equations is a concise, 1-term introduction to partial differential equations for the upper-level undergraduate/graduate course in Mathematics, Engineering and Science. Divided into two accessible parts, the first half of the text presents first-order differential equations while the later half is devoted to the study of second-order partial differential equations. Numerous applications and exercises throughout allow students to test themselves on key material discussed.
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Chapter 2 Second Order Partial Differential Equations
Fourier Transforms and Integrals
arbitrary constants arbitrary function auxiliary equations boundary conditions canonical form Cauchy problem characteristic curves characteristic strip compatible complete integral continuously differentiable functions d’Alembert’s solution deﬁne Diﬁerential Dirichlet problem domain eigenvalues envelope Example Exercise F(ac family of characteristic ﬁnd Find a complete Find the integral ﬁnding ﬁrst order p.d.e. ﬁve ﬁxed Fourier transform function of 90 g(ac given equation given p.d.e. Green’s harmonic function heat conduction Hence hyperbolic implies independent variables initial conditions initial curve initial data curve initial strip integral surface Jacobi’s method Laplace’s equation Lemma linear Monge cone Neumann problem non-linear Note Observe one-parameter family order partial differential ordinary differential equations parabolic parameter parametric equations partial differential equations Pfafﬁan differential equation Poisson integral formula quasi-linear equation respect to ac Riemann function satisﬁes satisfy solution of Equation Solve sub-family system of ordinary tangent plane two-parameter family u(ac unique wave equation