EFFICIENT NUMERICAL SOLUTIONS OF POISSON’S AND BIHARMONIC EQUATIONS

Abstract

Differential Equations are the language in which many laws of nature and their governing rules are expressed mathematically. Most physical phenomena can be modeled mathematically by second order and still some by fourth order partial differential equations; and these equations (PDEs) have become enormously successful as models of physical phenomena in all areas of engineering and sciences. The growing need for understanding the partial differential equations modeling of the physical problem has seen an increase in the use of mathematical theory and techniques, and has attracted the interest of many mathematicians. The elliptic type is perhaps one of the most important second order partial differential equation in applied mathematics. In engineering and many science fields one of the best known applicable theories in elliptic equations is potential equation. These equations describe many physical problems, like the slow motion of incompressible viscous fluid; the St. Venant theory of torsion; electrostatics; in heat and mass transfer theory; elasticity; magnetism and gravitating matter at points where the charge density, pole strength or mass density are non zero; and other areas of mechanics and physics. In particular, the Poisson’s equation describes stationary temperature distribution in the presence of thermal sources or sinks in the domain under consideration. In this thesis an attempt has been made to find efficient numerical solution of Poisson’s equation and biharmonic boundary value problem by considering different approximation schemes and extending the method of Hockney’s in Cartesian and cylindrical coordinate systems (including when r  0 is an interior or a boundary point ) with respect to the given boundary conditions. Chapter I is an introductory part and it deals with the important ideas and historical background of the development of finding the solution of Poisson’s equation. Chapter II deals with the numerical solution of the Poisson’s equation in a cube with the given Dirichlet’s boundary conditions. The Poisson’s equation is approximated by its equivalent finite difference second order approximation scheme in order to obtain a large number of algebraic ii linear equations and these equations are systematically arranged to get a block diagonal matrices structure. The obtained systems of block diagonal matrices are reduced, then, by extending the method of Hockney to a tri-diagonal matrix. Six examples have been considered in both cases and it is found that the method produce accurate results considering double precision. Chapter III deals with the numerical solution of the three dimensional Poisson’s equation approximated by a fourth order finite difference method in Cartesian coordinate systems in a cube with the Dirichlet’s boundary conditions. Based on the approximation scheme we have developed 19 and 27 points stencil schemes. Both schemes result in a large algebraic system of linear equations and are treated systematically in order to get a block tri-diagonal system by extending the method of Hockney, and these systems of linear equations are solved by the use of Thomas algorithm. It is shown that the method produce accurate results and moreover 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula. Six examples are taken to show the accuracy of the method and it is shown that the method produces accurate results. Part of this chapter has been published in the American Journal of Computational Mathematics 2011, Vol 1, No. 4 pp. 285-293. Chapter IV deals with the numerical solution of the three dimensional Poisson’s equation in cylindrical coordinate systems for r  0approximated by a second order finite difference method in a cylinder or portion of cylinder with the Dirichlet’s boundary conditions. Based on the approximation scheme we have transformed the Poisson’s equation in to a large number of algebraic systems of linear equations and these systems of linear equations are treated systematically in order to get a block tri-diagonal system, and these systems of linear equations are solved by the use of Thomas algorithm. Seven examples have been tested to verify the efficiency of the method and it is shown that this method produces good result. Part of this work is to appear in the American Journal of Computational Mathematics. Chapter V deals with the fourth-order numerical solution of the three dimensional Poisson’s equation in cylindrical coordinate systems for r  0with the Dirichlet’s boundary conditions. The Poisson’s equation is approximated by a fourth order finite difference approximation (19 points stencil scheme) to convert the equation in to a large number of system of algebraic linear iii equations; and the resulting large number of these algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. These systems of linear equations are solved by the use of Thomas algorithm, and using backward substitution we obtain the solution for the Poisson’s equation. Seven examples have been considered and it is shown that this method produces good result. Part of this work is to appear in the American Journal of Computational Mathematics. Chapter VI deals with the second and fourth-order approximation scheme for the numerical solution of the three dimensional Poisson’s equation in cylindrical coordinate systems when r  0 is an interior or a boundary point. Chapter VII deals with the numerical solution of the two (three) dimensional biharmonic boundary value problem of the second kind in a rectangular region (a cube) respectively, in Cartesian coordinate systems. Using the splitting method the two/three dimensional linear biharmonic boundary value problem is replaced by a coupled Poisson’s equations and these coupled Poisson’s equations are solved directly by using the fourth order finite difference approximation scheme which we have developed in Chapter III. For non-linear biharmonic boundary value problem of the second kind, we use splitting and iterative method together. Eight examples have been considered to test the efficiency of the methods. Finally, in Chapter VIII, based on the present study, conclusions are drawn and in this direction future research work is suggested.