Bernstein series solution of the heat equation in 2-D

A broad class of steady-state physical problems can be reduced to finding the harmonic functions that satisfy certain boundary conditions. The Dirichlet problem for the Laplace equation is one of the above mentioned problems. In this paper, a numerical matrix method is developed for numerically solving the Heat equation. The method converts the heat equation to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis is included to demonstrate the validity and applicability of the technique. Finally, the effectiveness of the method is illustrated in the heat equation for a cut ring region

Analytic Solution for Two-Dimensional Heat Equation for an Ellipse Region

In this study, an altenative method is presented for the solution of two-dimensional heat equation in an ellipse region. In this method, the solution function of the problem is based on the Green, and therefore on elliptic functions. To do this, it is made use of the basic consepts associated with elliptic integrals, conformal mappings and Green functions

LUCAS POLYNOMIAL SOLUTION FOR NEUTRAL DIFFERENTIAL EQUATIONS WITH PROPORTIONAL DELAYS

This paper proposes a combined operational matrix approach based on Lucas and Taylor polynomials for the solution of neutral type differential equations with proportional delays. The advantage of the proposed method is the ease of its application. The method facilitates the solution of the given problem by reducing it to a matrix equation. Illustrative examples are validated by means of absolute errors. Residual error estimation is presented to improve the solutions. Presented in graphs and tables the results are compared with the existing methods in literature