By Andrei D. Polyanin

It's a very infrequent celebration while a nonlinear partial differential equation admits a precise resolution. Such circumstances are of the maximum value as they permit the entire and such a lot particular research of the matter involved. This reference publication comprises the main prolonged checklist of nonlinear PDEs recognized to be solvable up to now and provides not just their special options but in addition answer methods.It is a very stable booklet.

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3 with f (t) = btn . 3. ∂x2 w + bw ∂w ∂t t ∂x Modified Burgers equation. =a ∂2w ∂x2 . 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = C1 w(C1 x + C2 , C12 t), w2 = w(x − bC3 t1−k , t) + C3 (1 − k)t−k w3 = w(x − bC3 ln |t|, t) + C3 t −1 if k ≠ 1, if k = 1, where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. © 2004 by Chapman & Hall/CRC Page 13 14 PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 2◦ . Degenerate solution linear in x: (1 − k)x + C1 C2 tk + bt x + C1 w(x, t) = t(C2 + b ln |t|) where C1 and C2 are arbitrary constants.

Kurdyumov, and A. P. Mikhailov (1995). 2 . 4. ξ= w(x, t) = 1 tan(λx), λ λ= − 1/2 b 3a , 3◦ . Multiplicative separable solution: w(x, t) = (t + C)3/4 u(x), where C is an arbitrary constant, and the function u = u(x) is determined by the autonomous ordinary differential equation a(u−4/3 ux )x + bu−1/3 − 34 u = 0. 4◦ . 6 with b = c = 0. ∂ ∂w =a w–3/2 + bw5/2 . ∂t ∂x ∂x 1◦ . Functional separable solution: 5. ∂w w(x, t) = a2/3 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t) Here, f2 (t) = 3 ϕ(t) dt + 3B, f1 (t) = 1 A ϕ(t) dt + B 3 1 1 ϕ(t) dt + B + ϕ(t) 9A2 6A2 where the function ϕ(t) is defined implicitly by f0 (t) = 2 −2/3 + .

3◦ . Generalized traveling-wave solutions: w = w(z), z = ♦ x + Cebt , where C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation a(wn wz )z − bzwz = 0. 1. EQUATIONS WITH POWER-LAW NONLINEARITIES ∂w ∂ ∂w + aw =b wn . ∂t ∂x ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function ∂w 16. w1 = C1 w(C11−n x + C2 , C12−n t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: wn dw = x + λt + C2 , aw2 + 2λw + C1 2b where C1 , C2 , and λ are arbitrary constants.