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By Lokenath Debnath
The revised and enlarged 3rd version of this profitable booklet offers a accomplished and systematic therapy of linear and nonlinear partial differential equations and their assorted and up to date purposes. with the intention to make the ebook extra helpful for a various readership, up to date smooth examples of functions are selected from parts of fluid dynamics, fuel dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation.
Nonlinear Partial Differential Equations for Scientists and Engineers, 3rd Edition, improves on an already hugely whole and obtainable source for graduate scholars and execs in arithmetic, physics, technology, and engineering. it can be used to nice influence as a direction textbook, learn reference, or self-study guide.
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Additional info for Nonlinear Partial Differential Equations for Scientists and Engineers
Example text
74) where amn are constants to be determined from the initial condition so that ∞ ∞ mπx a amn sin f (x, y) = u(x, y, 0) = m=1 n=1 sin nπy . 75) by sin( rπx a ) and integrate the result with respect to x from 0 to a with fixed y, so that ∞ a nπy a amn sin 2 n=1 b = mπx a f (x, y) sin 0 dx. 77) dx = g(y). 78) dy, whence amn = 4 ab a b f (x, y) sin 0 0 mπx a sin nπy b dx dy. 79). The method of construction of the solution shows that the initial and boundary conditions are satisfied by the solution.
B 2 − 4AC > 0. Equations for which B 2 − 4AC > 0 are called hyperbolic. 10ab) gives two real and distinct families of characteristics φ(x, y) = C1 and ψ(x, y) = C2 , where C1 and C2 are constants of integration. 3) reduces to the form 1 uξη = − ∗ D ∗ uξ + E ∗ uη + F ∗ u − G∗ = H1 (say). 11) B This is called the first canonical form of the hyperbolic equation. 12ab) are introduced, then uη = uα αη + uβ βη = uα − uβ , uξ = uα αξ + uβ βξ = uα + uβ , (uη )ξ = (uη )α αξ + (uη )β βξ = (uα − uβ )α · 1 + (uα − uβ )β · 1 = uαα − uββ .
29) becomes −4c2 uξη = 0. 33) where φ and ψ are arbitrary functions. Thus, in terms of the original variables, we obtain u(x, t) = φ(x − ct) + ψ(x + ct). 34) This represents the general solution provided φ and ψ are arbitrary but twice differentiable functions. The first term φ(x − ct) represents a wave (or disturbance) traveling to the right with constant speed c. Similarly, ψ(x + ct) represents a wave moving to the left with constant speed c. Thus, the general solution u(x, t) is a linear superposition of two such waves.