This thesis is concerned with the efficient numerical solution of nonlinear partial differential equations (PDEs) of elliptic and parabolic type. Such PDEs arise frequently in models used to describe many physical phenomena, from the diffusion of a toxin in soil to the flow of viscous fluids. The main focus of this research is to better understand the implementation and performance of nonlinear...

For simplicity, we adopt the following rules: x, x0, y, y0, z, z0, r denote real numbers, u, u0 denote elements of R3, f , f1, f2 denote partial functions from R3 to R, R denotes a rest, and L denotes a linear function. Let f be a partial function from R3 to R and let u be an element of R3. We say that f is partial differentiable on 1st-1st coordinate in u if and only if the condition (Def. 1) ...

In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate va...

In this paper, we show that for several second-order partial differential equations L[u] = A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy +D(x, y)ux + E(x, y)uy = λnu which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas ...

In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Let us note that every rest is total. Let f be a partial function from R2 to R and let z be an element of R2. The functor pdiff1(f, z) yielding a function from R2 into R is defined as follows: (Def....