NTMSCI 3, No. 4, 76-82 (2015) 76 New Trends in Mathematical Sciences http://www.ntmsci.com Solution of the two-dimensional heat equation for a rectangular plate Emel Kurul 1 and Nurcan Baykus Savasaneril 2 1 Department of Mathematics, Faculty of Sciences, Selcuk University, Konya, Turkey 2 Izmir Vocational School, Dokuz Eylul University, Izmir, Turkey Received: 4 March 2015, Revised: 23 March 2015, Accepted: 1 April 2015 Published online: 14 November 2015 Abstract: Laplace equation is a fundamental equation of applied mathematics. Important phenomena in engineering and physics, such as steady-state temperature distribution, electrostatic potential and fluid flow, are modeled by means of this equation. The Laplace equation which satisfies boundary values is known as the Dirichlet problem. The solutions to the Dirichlet problem form one of the most celebrated topics in the area of applied mathematics. In this study, a novel method is presented for the solution of two-dimensional heat equation for a rectangular plate. In this alternative method, the solution function of the problem is based on the Green function, and therefore on elliptic functions. Keywords: Heat equation, Dirichlet problem, elliptic functions, elliptic integral, green function. 1 Introduction Laplace’s equation is one of the most significant equations in physics. It is the solution to problems in a wide variety of fields including thermodynamics and electrodynamics. Today, the theory of complex variables is used to solve problems of heat flow, fluid mechanics, aerodynamics, electromagnetic theory and practically every other field of science and engineering. 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. The Dirichlet problem is to find a function U (z)that is harmonic in a bounded domain D ⊂ R 2 , is continuous up to the boundary ∂ Dof D, assumes the specified values U 0 (z) on the boundary ∂ D, where U 0 (z) is a continuous function on ∂ D, and can be formulated as ∇ 2 U = 0 , z ∈ D , U | z∈∂ D = U 0 (z) (1) Here, for a point (x, y) in the plane R 2 , one takes the complex notation z = x + iy, U (z)= U (x, y) and U 0 (z)= U 0 (x, y) are real functions and ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 is the Laplace operator. Similarly the Dirichlet problem for the Poisson equation can be formulated as ∇ 2 U = h(z) , z ∈ D , U | z ∈ ∂ D = U 0 (z) (2) Brovar et al. [1] investigated the relation between the Dirichlet problem and the Cauchy problem. Sezer [2] developed a new method for the solution of Dirichlet problem. Lanzara [3] studied Dirichlet problem for second degree elliptic linear c ⃝ 2015 BISKA Bilisim Technology ∗ Corresponding author e-mail: emelkurul@gmail.com: nurcan.savasaneril@deu.edu.tr