Classification of second order partial differential equation using maple and comparison for the solutions

The study of partial differential equations plays a significant role in many fields including mathematics, physics, and engineering. A partial differential equation (PDE) relates the partial derivatives of a function of two or more independent variables together. The general linear second order partial differential equation with constant coefficient has the form aZx x + bZx y +cZy y +dZx +eZy + f Z = g(x, y). There are many methods of solving this type of PDE’s such as finite elements, finite different and crank Nicolson depends on its classifications based on Z = b2-4ac. It is also well known that PDE is hyperbolic when -> 0, parabolic when - 0 and elliptic when-<0. In this research study, classification of the partial differential equation with constant coefficient is achieved by using Maple program. The classifications of variable coefficients of partial differential equations by Maple program are also given. The PDE’s after the convolution has the form AZx x + BZx y + CZy y + DZx + EZy + F Z = G, where A, B and C are coefficients of the PDE’s after the convolution. Further more the classification PDE’s with convolution are addressed by using -1=B2-4AC. Similarities in the classification of PDE’s before and after convolution were found. The solution of some important problems such as the wave equation is highly need of and occurs as one of three fundamental equations in mathematical physics that occurs in many branches of physics, applied mathematics, and engineering. In this research work, some problems of PDE’s with constant coefficient are solved by double Laplace transforms method. The same problems of the PDE’s are modified by some convolution function. The solutions of this new PDE’s are obtained by double Laplace transform. Graphical comparisons indicated that the methods are the same. In the same way, the PDE’s with variable coefficient are solved by double Laplace transforms methods. Then the same problem of the PDE’s is modified by some convolution functions. The new PDE’s are solved after convolution. However, graphical comparison made revealed that this two PDE’s before and after convolution are the same.

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