## Solving Nonlinear Partial Differential Equations with Maple by Inna Shingareva, Carlos Lizárraga-Celaya (auth.)

By Inna Shingareva, Carlos Lizárraga-Celaya (auth.)

The emphasis of this paintings is on developing forms of suggestions (exact, approximate analytical, numerical, graphical) of various nonlinear PDEs thoroughly, simply, and quick. The reader can examine a wide selection of options and remedy a variety of nonlinear PDEs incorporated and lots of different differential equations, simplifying and reworking the equations and recommendations, arbitrary services and parameters, offered within the book). various comparisons and relationships among numerous kinds of ideas, various tools and methods are supplied, the implications received in Maple and Mathematica, allows a deeper knowing of the subject

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**Additional resources for Solving Nonlinear Partial Differential Equations with Maple and Mathematica **

**Example text**

Applying the Maple predeﬁned functions pdsolve with option HINT=strip, dsolve, and charstrip, verify that the solutions (Sol1, Sol2), obtained by the method of characteristics read in the Maple notation: 24 Introduction Sol1 := {u ( s) = C2 , x ( s) = C2 s + C1 , y ( s) = − s + C3 } Sol2 := ux 2 + uy = 0 &where [{{u ( s) = − C3 s + C2 , x ( s) = 2 C4 s + C1 , y ( s) = s + C5 , p 1 ( s) = C4 , p 2 ( s) = C3 }} , { p 1 = ux , p 2 = uy }] Maple : with(PDEtools); declare(u(x,y)); PDE1:=u(x,y)*diff(u(x,y),x)=diff(u(x,y),y); PDE2:=diff(u(x,y),x)^2+diff(u(x,y),y)=0; sysCh:=charstrip(PDE1,u(x,y)); funcs:=indets(sysCh,Function); Sol1:=dsolve(sysCh,funcs,explicit); Sol2:=pdsolve(PDE2,HINT=strip); For the given equation PDE1, we ﬁrst obtain the characteristic system (depending on a parameter s) via charstrip, and solve this system via dsolve to obtain the solution Sol1 in the parametric form.

3 KdV equation. Nonlinear heat equation. Nonlinear telegraph system. Translation transformations. Let us consider the KdV equation (Eq11), the nonlinear heat equation (Eq21), and the nonlinear telegraph system (sys1): ut +uux +uxxx =0, ut − F (u(x, t))ux vt −F (u(x, t))ux −G(u(x, t))=0, x =0, ut −vx =0, where {x ∈ R, t ≥ 0}, (x, t) are the independent variables, (u, v) are the dependent variables, and F (u), G(u) are arbitrary functions. , the resulting equations and system take the form (Eq12, Eq22, sys2): uT +u(X,T )uX +uXXX , uT − F (u(X,T ))uX vT −F u(X,T ) uX −G(u(X,T ))=0, X =0, uT −vX =0.

Solution Sol20 is equivalent to pdsolve(Eq2,build). With the aid of the predeﬁned function TWSolutions we can ﬁnd diﬀerent types of traveling wave solutions (of nonlinear PDEs and their systems) according to a list of functions Fs (instead of the tanh function, by default). The algorithm is based on the tanh-function method and its various extensions (see Sect. 2). , all particular cases of more general solutions being constructed), we can obtain numerous traveling wave solutions (Sol21). For example, specifying the cot and tan functions and other options, we can obtain various traveling wave solutions.