Exact Solutions to Nonlinear Equations and Systems

Exact Solutions to Nonlinear Equations and Systems of Equations of General Form in Mathematical Physics A. D. Polyanin, A. I. Zhurov, and E. A. Vyazmina Citation: AIP Conference Proceedings 1067, 64 (2008); doi: 10.1063/1.3030831 View online: http://dx.doi.org/10.1063/1.3030831 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1067?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Amplitude‐Phase Formulation for a Discrete Two Species Lotka‐Volterra System AIP Conf. Proc. 1186, 451 (2009); 10.1063/1.3265360 Analytic Solutions of some Generalized Burgers Equations—An Overview AIP Conf. Proc. 1022, 17 (2008); 10.1063/1.2956179 Concerning an Analytical Solution of Some Families of Nonlinear Functional Equations AIP Conf. Proc. 936, 412 (2007); 10.1063/1.2790165 Nonlinear diffusion equation, Tsallis formalism and exact solutions J. Math. Phys. 46, 123303 (2005); 10.1063/1.2142838 Exact Evaluation of Collision Integrals for the Nonlinear Boltzmann Equation AIP Conf. Proc. 663, 35 (2003); 10.1063/1.1581522

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 47.19.107.202 On: Tue, 28 Apr 2015 12:41:50

Exact Solutions to Nonlinear Equations and Systems of Equations of General Form in Mathematical Physics A. D. Polyanin*, A. I. Zhurov^ and E. A. Vyazmina** * Institute for Problems in Mechanics, Russian Academy of Sciences 101 Vemadsky Ave., bldg, 119526 Moscow, Russia ^ Cardiff University, School of Dentistry, Heath Park, Cardiff CF14 4XY, UK **Ecole Polytechnique, 91128 Palaiseau Cedex, France Abstract. The paper gives an overview of recent results in exact solutions to nonlinear equations and system of equations of general form with functional arbitrariness. Special attention is paid to equations that arise in heat and mass transfer, wave theory, and mathematical biology. The solutions considered below have been obtained, for the most part, with the method of generaUzed separation of variables and that of functional separation of variables. Some of the solutions presented are new. Keywords: Exact solutions, generalized separable solutions, functional separable solutions, nonlinear equations, systems of equations, mathematical physics. PACS: 02.30.Jr

STRUCTURE OF GENERALIZED AND FUNCTIONAL SEPARABLE SOLUTIONS To simphfy the presentation, we confine ourselves to the case of mathematical physics equations in two independent variables, x and t, and a dependent variable, w. Definition 1. Exact solutions to nonlinear PDEs in the form w{x,t) = q)i{x)Yi{t) + q>2{x)W2{t) + ••• + q)n{x)Wn{t)

(1)

are called generalized separable solutions. Definition 2. Exact solutions to nonlinear PDEs in the form w{x,t)=F{z),

z=q)i{x)Yi{t)

+ q>2{x)W2{t) + --- + q>n{x)Yn{t)

(2)

are called functional separable solutions. Substituting (1) (or (2)) into the equations under consideration leads to functionaldifferential equations for (pi{x) and v/^(?) (and F(z)). These functional-differential equations may be solved using the method of differentiation and the splitting method, which are described in the books [1-3] (see also the Titov-Galaktionov method [4-6], based on using invariant subspaces). Presented below are original nonlinear equations and respective final exact solutions, without intermediate results.

CPIOGI, Applications of Mathematics in Engineering and Economics '34—AMEE '08, edited by M. D. Todorov O 2008 American Institute of Physics 978-0-7354-0598-l/08/$23.00

64 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 47.19.107.202 On: Tue, 28 Apr 2015 12:41:50

GENERALIZED POROUS MEDIUM EQUATION WITH A NONLINEAR SOURCE Consider the following mass transfer equation for a medium with complex rheology and a power Pick's law for the concentration gradient: dw dt

d dx

/'"''(S)"

-8{w)

(3)

This is a generalized porous medium equation with nonlinear source [7]. In the special case « = 1, equation (3) is the ordinary diffusion (heat) equation with volume reaction (heat source), whose invariant solutions are described in [8]. Generalized and functional separable exact solutions to equation (3) with can be found in [1, 3, 9]. In the general case, equation (3) admits an obvious traveling-wave solution w = w{z), z = x + Xt, which will not be considered. Given below are more complicated exact solutions obtained in [10, 11]. 1°. Suppose the function / = /(w) is arbitrary and the function g = g{w) is defined as g{w)=A[f{w)]-'/"-B, where A and B are some numbers. In this case, equation (3) admits a functional separable solution, which is defined imphcitly by J

B[n+l)

where Ci and C2 are arbitrary constants. 2°. lfg = g{w) is arbitrary and / = /(w) is defined as

/w =[g{w)Y exp Bn

dw 8{w)

where A and B are some numbers, then equation (3) admits a functional separable solution given implicitly by dw = t g{w)

1 ln\Bx+Ci\ + C2. B

3°. If ^ = g(w) is arbitrary and / = f(w) is defined as /(w)

=

AiA^w + B

A^Ai

b(w)]"

fe(w)]"

Z = A:

dw

g{wy

Zdw,

(4) (5)

65 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 47.19.107.202 On: Tue, 28 Apr 2015 12:41:50

where Ai, A2, and A3 are some numbers, then equation (3) admits generalized travelingwave solutions of the form w = w{Z),

Z=(p{t)x+ii/{t),

where the function w{Z) is determined by the inversion of (5), while (p{t) and \j/{t) are expressed as 1 Ci-A3{n+l)t

(p{t)

(p{t)

W{t)

dt

[(p{t)fdt+A2

W)

C2

with Ci and C2 being arbitrary constants. 4°. If ^ = g(w) is arbitrary and / = /(w) is defined as 1

/(w)

ji A1W+A3 / Zdw

M^)r

Z = -— exp A4 A4

dw g{w)

exp MA4

dw g{w)

A2 "A4'

(6) (7)

where Ai, A2, A3, and A4 are some numbers (A4 ^ 0), then equation (3) admits generalized traveling-wave solutions of the form w = w{Z),

Z=(p{t)x+ii/{t),

where the function w{Z) is determined by the inversion of (5), while (p{t) and \j/{t) are expressed as (p{t) = ¥{t)

Cie

= -

^

f»

{x)-bi\if{t)], w = \\ai^r{t) -a2(p{x)], A = aib2 - a2bi / O

F = uf{w), G = u'^giw) F = f{au + bw), G = g{au + bw)

R

+

•^ '^

Right-hand sides of (15)

^

1 R

+

F = f{a\u + b\w), G = g(a2U + b2w) F = f{au — bw), G = ug{au — bw) + h{au — bw)

u= (p{t) + b0{t)x, w= \i/(t) +a0(t)x

OQ

a R

+

F = f{au — bw) + cw, G = ug{au — bw) + h{au — bw) F = aulnu + uf{u"w^), G = wg{u"w^) F = uf{au" + bw), G = u'^giau'^ + bw)

u=(p{t)+be{t)e^'', w= Yit) + aO(t)e'^'', X = ac/b u = exp(Cme'"))'(^), w = exp(—Cne'")z(^), ^ = kx — Xt 1

M = (Cii + C2)«-*e(x), w = (p(x)-f(Cif+C2)^[e(x)]"

Determining equations v'z = f{w)v, W^ = g(M')v*

V'z = /(»*')V,

w[ = g(M')v* k2y'^+bki=f(ay + bz), -.^24 +a-h = gii^y + bz) ^2