1), find the explicit fundamental solution to the heat equation ∆u + b · ∇u − u t = 0 in R n × (0, ∞). (1) Letting G be what you find, show u 0 (x) = lim t→0 + R n G(x, t; y, 0)u 0 (y) dy, (2) when u 0 is bounded and continuous on R n. Taking the Fourier transform (with respect to x) of the equation gives −|ξ| 2 ˆ u + ib · ξ ˆ u − ˆ u t = 0 − |ξ| 2 + ib · ξ ˆ u = ˆ u t =⇒û = ce −(|ξ| 2 +ib·ξ)t ...

Evolution equations associated with irreversible physical processes like diffusion and heat conduction lead to parabolic partial differential equations. When the equation is a model for a reversible physical process like propagation of acoustic or electromagnetic waves, then the evolution equation is generally hyperbolic. The mathematical models usually begin with a conservation statement that ...

the homogeneous balance method can be used to construct exact traveling wave solutions of nonlinear partial differential equations. in this paper, this method is used to construct newsoliton solutions of the (3+1) jimbo--miwa equation.

in this paper, the solution of the evolutionaryfourth-order in space, sivashinsky equation is obtained by meansof homotopy perturbation method (textbf{hpm}). the results revealthat the method is very effective, convenient  and quite accurateto systems of nonlinear partial differential equations.

COURSE DESCRIPTION The course is an introduction to the study of partial differential equations (PDEs) using functional analysis and energy methods. Questions of existence, uniqueness and regularity for weak solutions to linear elliptic and parabolic PDEs will be emphasized. Various nonlinear PDEs will also be studied, using a variety of different approaches, like variational and monotonicity m...

 In this paper, we present "a posteriori" analysis of the fundamental concepts involved in the modelling of problems of mathematical physics by Partial Differential Equations (PDEs). Our aim is to improve our students' understanding of PDEs when applied to an engineering problem, from a completely qualitative point of view. They should be able to understand the deep meaning of any Laplacian, c...

A different approach is developed to study self-similar solutions in Bianchi type III spacetimes. Self-similar solutions in both tilted and non tilted cases are obtained. By solving a system of six non linear and inhomogeneous partial differential equations. It is shown that in non-tilted case the above space-times admit orthogonal zero kind self-similarity and in tilted case. It admits second ...

The ultimate topic to be touched on in this book is the vast and active field of nonlinear partial differential equations. Leaving aside quantum mechanics, which remains to date an inherently linear theory, most real-world physical systems, including gas dynamics, fluid mechanics, elasticity, relativity, ecology, neurology, thermodynamics, and many more, are modeled by nonlinear partial differe...