Numerical Solution of Partial Differential Equation Finite Random Grids
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International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 12, Issue 4, April 2021, pp.34-45 Article ID: IJARET_12_04_005 Available online at http://iaeme.com/Home/issue/IJARET?Volume=12&Issue=4 ISSN Print: 0976-6480 and ISSN Online: 0976-6499 DOI: 10.34218/IJARET.12.4.2021.005 © IAEME Publication Scopus Indexed NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATION FINITE RANDOM GRIDS Sanaullah Mastoi* Quaid-e-Awam University of Engineering Science and Technology (Campus), Larkana 77150, Sindh, Pakistan; Institute of Mathematical Science, University of Malaya, Kuala Lumpur 50603, Malaysia Nabi Bux kalhoro Quaid-e-Awam University of Engineering Science and Technology (Campus), Larkana 77150, Sindh, Pakistan. Ali Bakhsh Mugheri Quaid-e-Awam University of Engineering Science and Technology (Campus), Larkana 77150, Sindh, Pakistan. Arifa Bano Talpur Mehran University Institute of Science, Technology and Development (MUISTD), Mehran University of Engineering and Technology (MUET), Jamshoro 76060, Sindh, Pakistan Umair Ahmed Rajput Quaid-e-Awam University of Engineering Science and Technology, Nawabshah (67480), Pakistan Ruqia Bano Mastoi Department of English, Faculty of Social Sciences, Shaheed Benazir Bhutto University, Shaheed Benazirabad 67450, Sindh, Pakistan Niamtullah Mastoi Mehran University Institute of Science, Technology and Development (MUISTD), Mehran University of Engineering and Technology (MUET), Jamshoro 76060, Sindh, Pakistan Wan Ainun Binti Mior Othman* Institute of Mathematical Science, University of Malaya, Kuala Lumpur 50603, Malaysia. *Corresponding Author http://iaeme.com/Home/journal/IJARET 34 [email protected] Sanaullah Mastoi, Nabi Bux kalhoro, Ali Bakhsh Mugheri, Arifa Bano talpur, Umair Ahmed Rajput, Ruqia Bano Mastoi, Niamtullah Mastoi and Wan Ainun Binti Mior Othman ABSTRACT In this paper, numerical solution of second-order basic partial differential equations is solved and compared. Various method used in solving second order partial differential equation. Numerical solution using finite difference method over (Uniform Vs Randomly) meshes Vs Theta Method are presented and discussed. To examine the practicability of PDEs through finite difference method using randomly generated grids. Numerical Solution using Finite difference method is based on grids. The idea of randomly generated meshes helps to decide the practicability and feasibility of such approaches. In this research, time, iterations, and performance measured. We have solutions through randomly generated grids having better results than uniform meshes and theta method. Key words: Finite difference method, fractional differential equations, Randomly generated meshes, Uniform meshes, Theta method. Cite this Article: Sanaullah Mastoi, Nabi Bux kalhoro, Ali Bakhsh Mugheri, Arifa Bano talpur, Umair Ahmed Rajput, Ruqia Bano Mastoi, Niamtullah Mastoi and Wan Ainun Binti Mior Othman, Numerical Solution of Partial Differential Equation Finite Random Grids, International Journal of Advanced Research in Engineering and Technology (IJARET), 12(4), 2021, pp. 34-45. http://iaeme.com/Home/issue/IJARET?Volume=12&Issue=4 1. INTRODUCTION Consider the Partial differential equations is Poisson's equation. which is an elliptic type of Partial Differential Equation (PDE) which is widely used to simulate the various physical processes [1, 2]. To obtain the numerical solution of Poisson's equation the proper choice of mesh (alternatively called the grid) is the foremost step to achieve the better accuracy and convergence. However, the mesh generation process is not unique since there are no universal rules (formulas) to be discovered. Usually, the meshes are designed according to the problem and physical structures. It is often found that the better mesh quality led to the greater rate of convergence [3]. But the design of the most feasible mesh for a particular problem is challenging due to the variability in the boundary conditions and the structure of domain. A given 2D domain is filled with structured or unstructured meshes with quadrilateral or triangular elements, respectively. The structured meshes are naturally easier to compute and implement and may require more elements or worse-shaped elements whereas, the unstructured meshes are often computed by Delaunay triangulation of point sets [4]. There are quite varied approaches for structured and unstructured meshes having their own merits and limitation. The motivation of this research is based on the idea of using randomly generated meshes to solve Poisson’s equation by finite difference method. Therefore, it is hypothesized that “the randomly generated meshes may improve the convergence of numerical solution.” If the numerical convergence is improved, then better accuracy can be obtained from the randomly generated meshes. The term mesh generation or alternatively the grid generation is process of generating polygonal or polyhedral mesh that approximate a geometric domain (Wikipedia). Different types of grids like Cartesian grid, triangular grids, curvilinear grid, rectilinear grid and many others are used to approximate the domain, but all types of grids can be put into two categories that are structured and unstructured. Structured meshes is characterized by (uniform) regular connectivity and normally use quadrilateral elements in 2D while an unstructured mesh is characterized by (non-uniform) irregular connectivity and uses triangular elements. A scheme for combining structured and unstructured meshes, in single approach, the beneficial properties http://iaeme.com/Home/journal/IJARET 35 [email protected] Numerical Solution of Partial Differential Equation Finite Random Grids offered for these two mesh types were discussed [5]. The meshes are designed according to the problem and physical structures. With the advent of computer technology, a vast variety of numerical algorithms have been proposed and developed to obtain the solution of partial differential equations (PDEs) with high speed of convergence and accuracy. However, the development of new numerical algorithms for PDEs inherently depends on the way how the mesh is constructed [6]. The mesh generation has gained much attention due to its applicability to all physical problems particularly, in computational fluid dynamics, electromagnetism and structural mechanics. Numerically generated meshes have offered the key to removing the problem of boundary shape from finite difference methods, and these grids also can serve for the construction of finite element meshes [7]. The numerical solution of different types of PDEs by a variety of discretization techniques has been proposed in literature. However, the solution strategy varies with the physical phenomenon to be simulated, the type of governing equation and the computational domain of the problem. Initially, the studies have been done on regular meshes with equal step size [8]. But few studies can also be found on variable grid size as can be seen in the work of [9, 10]. The concept of moving meshes [11,12] for quasi linear PDEs have also been applied to solve the problems in magneto hydrodynamics [13], meteorology [14], porous medium [12], free surface viscoelastic flow [15] multiphase flows [16] and low speed viscous flow [17], etc. Similarly, the notion of strand meshes has applied by some researchers [18, 19] to solve complex flow fields and aerodynamics problems. In summary, the mesh based finite element and boundary elements methods were developed in the final two decades of last century with regular or irregular generation of grids to solve many scientific and engineering problems [20, 21]. The random grids or non-uniform or unstructured grids are based on the scattered points in which partial differential equation is solved as a means for local smoothing. Classical finite difference schemes are naturally based on regular grids, and therefore problem domain is restricted with regular geometry [22]. The random meshes are often best suitable for the finite element method but for the finite difference method [23, 24] (Olof Runborg, 2012), but it is difficult to decide for finite difference method since there is no appropriate data available. Generated grid size is rarely found in the literature. Therefore, the investigation of the effect of random meshes on the performance of FDM is the main motivation of the proposed research. For testing and implementation purpose, a 2D Poisson's equation with Dirichlet boundary conditions will be considered the performance of finite difference method is based on the discretization schemes and the parameters of mesh. It is often observed that the FDM operates either on predefined equal grid size or variable grid size. Increase the validity and efficiency of Kac’s method using the random values in time, exponential distribution solved telegrapher’s equation by numerical method [25]. 2. DERIVATION OF POLYNOMIALS First, assuming the function whose derivatives are to be approximated is properly behaved, by Taylor's theorem, we can create a Taylor series expansion where 푛! denotes the factorial of n, and 푅푛(푥) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. We will derive an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial: http://iaeme.com/Home/journal/IJARET 36 [email protected] Sanaullah Mastoi, Nabi Bux kalhoro, Ali Bakhsh Mugheri, Arifa Bano talpur, Umair Ahmed Rajput, Ruqia Bano Mastoi, Niamtullah Mastoi and