Applied Mathematics and Physics, 2017, Vol. 5, No. 2, 40-46 Available online at http://pubs.sciepub.com/amp/5/2/2 ©Science and Education Publishing DOI:10.12691/amp-5-2-2 Calculation of Quantity Rate on the Rectangular Domain by Boundary Element Method Milad Bamdadinejad, Hassan Ghassemi * , Mohammad Javad Ketabdari Department of Maritime Engineering, Amirkabir University of Technology, Tehran, Iran *Corresponding author: gasemi@aut.ac.ir Abstract The purpose of the paper is to achieve relative error changes of influence coefficients based on the number of Gaussian points and relative error changes of   and   according to x and y using boundary element method (BEM) and constant elements. In this case, the dominant equation is Laplace’s equation defined for a rectangular domain with the Dirichlet boundary condition. The boundaries of the domain will first be discretization with four constant element and four boundary conditions will be introduce in MATLAB and then four Neumann boundary conditions will be gain. Afterwards, four influence coefficients have been obtained regarding the source point within the domain and first element analytical and numerical and their relative error has been computed. Finally,   and   values in four points toward x and three points toward y within the domain have been computed analytical and numerical and the results have been Presented in schemes and tables. Keywords: boundary element method, laplace equation, constant element, influence coefficients, relative error Cite This Article: Milad Bamdadinejad, Hassan Ghassemi, and Mohammad Javad Ketabdari, “Calculation of Quantity Rate on the Rectangular Domain by Boundary Element Method.” Applied Mathematics and Physics, vol. 5, no. 2 (2017): 40-46. doi: 10.12691/amp-5-2-2. 1. Introduction In mathematics, PDE is a differential equations involving functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multi-dimensional systems. PDEs find their generalization in stochastic partial differential equations [1]. Laplace's equation is the simplest form of elliptic PDEs. Due to simplicity and appropriate accuracy of Laplace's equation, it is used in different physical and engineering problem such as fluid dynamic [2], elasto-static [3], controls and vibrations [4] and heat transfer [5]. Potential theory is the general theory to solve of Laplace's equation. Moreover, solutions of Laplace's equation are harmonic functions. Many researches are addressed different method for solving Laplace’s equation under various boundary conditions [6,7,8,9]. For example, Laplace’s equation can be solved by separation of variables methods [10]. The Boundary Element Method (BEM) constitutes a technique for analyzing the behavior of mechanical systems and especially of engineering structures subjected to external loading. The term loading is used here in the general sense, referring to the external source which produces a non-zero field function that describes the response of the system (temperature field, displacement field, stress field, etc.), and it may be heat, surface tractions, body forces, or even non homogeneous boundary conditions, e. g. support settlement. Since its beginnings in the 1960s, the boundary element method (BEM) has become a well-established numerical technique which provides an efficient alternative to the finite difference and finite element method for solving a variety of engineering problems [11]. The classical BEM considered in this work requires a fundamental solution to the governing differential equation (here the Laplace equation) in order to obtain an equivalent boundary integral equation. Regarding homogeneous potential problems, BEMs have the following advantages [12]. The integral equation may be regarded as an exact solution of the governing partial differential equation. The boundary element method attempts to use the given boundary conditions to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post- processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain [13]. Types of element in BEM include variety of choices regarding order of the polynomial that defines them. Studying linear element is the first step in implementing higher order elements in BEM. It will be said that deriving the equations in linear element complies with the concept of constant element in many ways [14].