Download Handbook of Nonlinear Partial Differential Equations by Andrei D. Polyanin PDF
By Andrei D. Polyanin
It's a very infrequent get together while a nonlinear partial differential equation admits a precise resolution. Such circumstances are of the maximum significance as they permit the complete and so much specific research of the matter involved. This reference booklet includes the main prolonged record of nonlinear PDEs identified to be solvable so far and offers not just their distinct options but additionally answer methods.It is a very stable publication.
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Extra info for Handbook of Nonlinear Partial Differential Equations
Example text
T ∂x ∂x The substitution u = w −4/3 leads to an equation with quadratic nonlinearity: 6. ∂w 2 ∂ 2 u 3 ∂u 4 ∂u =u 2 − + au2 − cu − b . ∂t ∂x 4 ∂x 3 1◦ . For a = 1, there is a solution of the form u = ϕ1 (t) + ϕ2 (t) cos(kx) + ϕ3 (t) sin(kx) + ϕ4 (t) cos(2kx) + ϕ5 (t) sin(2kx), k = 2 × 3−1/2 , where the functions ϕn = ϕn (t) are determined by the system of first-order ordinary differential equations (not written out here). 1. EQUATIONS WITH POWER-LAW NONLINEARITIES 2◦ . For a = −1, there is a solution of the form ❛☎❜ u = ϕ1 (t) + ϕ2 (t) cosh(kx) + ϕ3 (t) sinh(kx) + ϕ4 (t) cosh(2kx) + ϕ5 (t) sinh(2kx), k = 2 × 3−1/2 .
Generalized self-similar solution: w = e−2λt ϕ(u), u = xeλmt , λ is any, where the function ϕ = ϕ(u) is determined by the ordinary differential equation a(ϕm ϕu )u = λmuϕu − 2λϕ. (6) This equation is homogeneous, and, hence, its order can be reduced (and then it can be transformed to an Abel equation of the second kind). The substitution Φ = ϕm+1 brings (6) to an equation that coincides, up to notation, with (5). 9◦ . Solution: w = (t + A)−1/m ψ(u), u = x + b ln(t + A), A, b are any, where the function ψ = ψ(u) is determined by the autonomous ordinary differential equation a(ψ m ψu )u = bψu − ψ/m.
Solutions: w(x, t) = Ax + B + bt, b x2 + Ax + B − (C − 2at), C − 2at 4a where A, B, and C are arbitrary constants. The first solution is degenerate and the second one is a generalized separable solution. w(x, t) = 3◦ . Traveling-wave solution: w = w(z), z = kx + λt, where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation ak 2 wwzz − λwz + b = 0. 4◦ . Self-similar solution: w = tU (ξ), ξ = x/t, where the function U (ξ) is determined by the autonomous ordinary differential equation aU Uξξ + ξUξ − U + b = 0.