Abstract
Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.
Highlights
Differential equations are mathematical expressions that how the variables and their derivatives with respect to one or more independent variables affect each other in dynamic way
A partial differential equation is a differential equation in which the unknown function F is a function of multiple independent variables and of their partial derivatives or Equations involving one or more partial derivatives of a function of two or more independent variables is called partial differential equations (PDEs)
A PDE is linear if the dependent variable and its functions are all of first order
Summary
Differential equations are mathematical expressions that how the variables and their derivatives with respect to one or more independent variables affect each other in dynamic way. This is the general second order, linear and nonhomogeneous partial differential equation. The partial differential equation is classified as parabolic, hyperbolic and elliptic depending on the values of A, B and C in the above equation (1). PDEs derived by applying a physical principle such as conservation of mass, momentum or energy. Pure and Applied Mathematics Journal 2016; 5(4): 120-129 mechanical behavior of general bodies are referred to as Conservation Laws. These laws can be written in either the strong of differential form or an integral form
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