The Decomposition Method in Several Dimensions

Abstract

Mathematical physics deals with physical phenomena by modelling the phenomena of interest, generally in the form of nonlinear partial differential equations. It then requires an effective analysis of the mathematical model, such that the processes of modelling and of analysis yield results in accordance with observation and experiment. By this, we mean that the mathematical solution must conform to physical reality, i.e., to the real world of physics. Therefore, we must be able to solve differential equations, in space and time, which may be nonlinear and often stochastic as well, without the concessions to tractability which have been customary both in graduate training and in research in physics and mathematics. Nonlinear partial differential equations are very difficult to solve analytically, so methods such as linearization, statistical linearization, perturbation, quasi-monochromatic approximations, white noise representation of actual stochastic processes, etc. have been customary resorts. Exact solutions in closed form are not a necessity. In fact, for the world of physics only a sufficiently accurate solution matters. All modelling is approximation, so finding an improved general method of analysis of models also contributes to allowing development of more sophisticated modelling [1, 2].