Solving nonlinear ODEs using Maple

Solving Nonlinear ODEs Using Maple s dsolve Command... 

In this chapter, nonlinear IVPs were solved numerically. In section 2.2.2 a noniinear IVP was solved analytically using Mapie s dsolve command. This approach is limited to very few nonlinear ODEs. In section 2.2.3, series solutions were obtained using Maple s dsolve command. This approach is valid for all... 

Nonlinear parabolic and elliptic partial differential equations are solved using the similarity solution technique in this section. The methods described in section 4.4 and sections 4.5 are valid for nonlinear partial differential equations, also. The methodology involves converting the governing equation (PDE) to an ordinary differential equation in the combined variable (Tj). This variable transformation is very difficult to do by hand. In this section, we will show how this variable transformation can be done using Maple. The original problem becomes a nonlinear boundary value problem (ODE) in the new combined variable (Tj). This is best illustrated with the following examples. 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

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