Analytical and Numerical Techniques for the Solution of Transient Partial Differential Equations

Abstract

Many different types of real-world physical phenomena arising in different fields of newlineengineering and sciences can be modeled by means of partial differential equations (PDEs), newlinefor example, aerodynamic flows, fluid flow, traffic flow problems, bio-maths, etc., and it is newlineimportant to study and solve such PDE s to further understand these physical processes. newlinePDEs can typically be classified up to the second order ( the highest derivative in the newlinePDE). There are three kinds of second-order PDEs, including parabolic, hyperbolic, and newlineelliptic. The solutions of hyperbolic equations are more fascinating than the solutions of newlineparabolic and elliptic equations. In general, elliptic PDE solutions are always smooth, newlinedespite the presence of singularities or sharp corners in the initial conditions. In most newlinecases, parabolic PDEs depend on the evolution of time, and the models include diffusion newlinecomponents. In hyperbolic partial differential equations, the smoothness of the solution is newlinedependent on the smoothness of the initial conditions in a linear advection problem. For newlineexample, if there is a discontinuity in the data at the beginning or at the boundary, then newlineperhaps the solution will have a discontinuity that is caused by the jump. Although both the newlineinitial conditions and the boundary conditions are smooth, the emergence of shocks is still newlinepossible in the case when the PDE possesses a nonlinearity property newline