Numerical Solution of Partial Differential Equation Finite Random Grids

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

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

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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:

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3. MATERIAL AND METHODOLOGY: 3.1 Theta Method Fractional Poisson Equations 휕훼푢(푥,푦) 휕훼푢(푥,푦) ∇훼푢(푥, 푦) = + = 푓(푥, 푦) on a finite domain [0,1] × [0,1] (1) 휕푥훼 휕푦훼 The entire model problem specifications are described and then the finite difference discretization of the governing equation is presented. After that the mesh generation process is defined for both uniform and random spacing. For the analysis of results, sufficient data is collected in terms of uniform and random meshes viz. ten different meshes of uniform spacing are generated and hundred other meshes with random spacing are generated.

Model Problem Specification Governing equation is chosen as two-dimension partial differential equation and applied on unit square with initial and boundary conditions as shown in the following figure.

휕2푢 휕2푢 + = 0, 0 ≤ 푥 ≤ 1, 0 ≤ 푦 ≤ 1, 휕푥2 휕푦2 푤푖푡ℎ 푢(푥, 푦) = 0 푢(0, 푦) = 푢(1, 푦) = 푎, 푢(푥, 0) = 푢(푥, 1) = 푏 (2)

where 푢 is dependent variable that depends on 푥 and y coordinates, where f is a given forcing function.

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Figure 1 Grid network a) Computation domain, b) closed view of the meshes.

3.2 Finite Difference Discretization of the Problem After the model problem specification, the governing equation is discretized by using the finite difference scheme. Since the problem is well-posed and does not require any special treatment regarding the consistency and stability. Therefore, the second order accurate central finite difference scheme is used, and the resulting system is described by the following Eq. 2

푢(푖+1,푗)−2푢(푖,푗)+푢(푖−1,푗) 푢(푖,푗+1)−2푢(푖,푗)+푢(푖,푗−1) + = −푓(푖, 푗) ℎ2 푘2 ∀푖,∈ℕ, 1 ≤ 푖 ≤ 푚, 1 ≤ 푗 ≤ 푛. 푤푖푡ℎ 푢(푖, 푗) = 0. (3) 푢(푖, 1) = 푎1, 푢(푖, 푛) = 푎2, 푢(1, 푗) = 푏1, 푢(푚, 푗) = 푏2

The computational domain is discretized accordingly as a finite difference mesh. A schematic of discretized mesh with uniform spacing is exhibited in Figure 2

Figure 2 A schematic of discretized domain

3.3 Generation of Uniform Finite Difference Meshes The uniform meshes are generated by writing a MATLAB code implemented on each interior node by Gauss Seidel iterative method explicitly by Eq.3 as follows in equation (4) 푘2(ℎ2 푓(푖,푗)+푢푝(푖+1,푗)−푢푝(푖−1,푗))+ℎ2(푢푝(푖,푗+1)−푢푝(푖,푗−1)) 푢푝+1(푖, 푗) = (4) 2(ℎ2+푘2) where 푝 stands for iteration and 푝 + 1 is the successive iteration.

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In order to solve the governing equation and obtain sufficient numerical results the 10 different sizes of meshes with equally spaced increments along x and y axes are generated (uniform meshes). The mesh data is shown in the following Table. 1,

Table 1 Data for uniform mesh generation S.No Mesh size Step size Number Number Average Cell standard (푚 × 푛) (푑푥, 푑푦) = (ℎ, 푘) of node of cells cell size deviation 1 10x10 0.1 121 100 0.1 0 2 20x20 0.05 441 400 0.05 0 3 30x30 0.033333 961 900 0.0333 0 4 40x40 0.025 1681 1600 0.025 0 5 50x50 0.02 2601 2500 0.02 0 6 60x60 0.0167 3721 3600 0.0167 0 7 70x70 0.0143 5041 4900 0.0143 0 8 80x80 0.0125 6561 6400 0.0125 0 9 90x90 0.0111 8281 8100 0.0111 0 10 100x100 0.01 10201 10000 0.01 0 The uniform meshes are generated by writing a MATLAB code and meshes constructed by using the data given by Table. 1 are two of them are shown in Figures 3 and 4.Then numerical solution is implemented on each interior node by Gauss Seidel iterative method.

Figure 3 Uniform mesh of size (10x10)

Figure 4 Uniform mesh of size (100x100)

3.4 Generation of Random Finite Difference Meshes As the generation of uniform meshes, the samples of random meshes are required to test their feasibility as mentioned in the objectives of this research. It can be observed from the data of

http://iaeme.com/Home/journal/IJARET 39 [email protected] Numerical Solution of Partial Differential Equation Finite Random Grids uniform meshes (Table. 1) that the mesh parameters like step size h, k, minimum cell size, maximum cell size and average cell size remain same for a particular grid size. However, when a random mesh is generated, all parameters change randomly not only for a specific grid size but also as many times the mesh is generated so many times the mesh parameters vary. For testing and implementation purpose, ten samples of each mesh size that is (10x10, 20x20, … 100x100) are obtained on MATLAB using the built-in function rand (given mesh size), it means for all ten different mesh sizes we have 10 different samples (or realizations) totaling a 100 meshes each one is different. The mesh parameters are further extended as minimum cell size, maximum cell size, average cell size, standard deviation of cells, skewness of cells, correlation in cells along x and y direction. These parameters are useful for choosing an appropriate numerical solution scheme and for the decision of selection of random samples with fast convergence. For uniform meshes the numerical solution can be implemented using the Eq. 3 but for random meshes the situation changes due to instantaneous change in the step size h and k as can be seen in Figure 5 & 6. 푘2 (ℎ2 푓(푖,푗)+푢푝(푖+1,푗)−푢(푖−1,푗))+ℎ2 (푢푝(푖,푗+1)−푢푝(푖,푗−1)) 푝+1 푚푎푥 푚푎푥 푚푎푥 푢 (푖, 푗) = 2 2 (5) 2(ℎ 푚푎푥+푘 푚푎푥)

Figure 5 Random spacing on a typical sample

3.5 Theta Method Definition 2.4.1 (Riemann-Liouville Fractional derivatives) if f be a real function and have continuous integer order r, then 푑푠푓(푥) 1 푑푟 푥 푓(훽) 퐷푠푓(푥) = = ∫ 푑훽 (6) 푥 푑푥푠 (푟−푠)! 푑푥푟 0 (푥−훽)푠+1−푟 is Riemann-Lioville fractional derivatives of order s that 푟 − 1 < 푠 < 푟. Definition 2.4.2 (Shifted Grunwald (SG)formula) for 1< s <2, the formula defines that 푑푠푓(푥) 1 푄 = lim ∑ 푔푠,푘.푓(푥 − (푘 − 1)ℎ, (7) 푑푥푠 푄→∞ ℎ푠 푘=0 as SG estimated for fractional derivatives, 푑푠푓(푥) 1 = ∑푄 푔 푓(푥 − (푘 − 1)ℎ + 푂(ℎ푠), (8) 푑푥푠 ℎ푠 푘=0 푠,푘. Q is a positive integer, more ever, theta methods is Definition 2.4.3 Theta method 1 푓 = (∑푖+1 푔 . 푈 + ∑푗+1 푔 . 푈 ) (9) 푖,푗 ℎ푠 푡=0 푠,푡 푖−푡+1,푗 푧=0 푠,푧 푗−푧+1

http://iaeme.com/Home/journal/IJARET 40 [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 4. SOLUTION OF PARTIAL DIFFERENTIAL EQUATION (UNIFORM VS RANDOM GRIDS VS THETA METHOD) Figures 7 shows that exact solution and 8 show the numerical simulation of steady state temperature in the unit square over uniform meshes of size 10x10, 20x20, …, 100x100 respectively. Figure exhibits local solution profile where the numerical solution values vary from 25 to 100. The smoothness in the solution verifies the well posedness and consistency of the problem which is the best properties of Poisson’s equation. Increasing the mesh size increases the smoothness. In fact, the purpose of obtaining these simulation profiles is to use as benchmarking for the simulation profiles obtained on random meshes. It will facilitate to make decision about the feasibility of using random meshes for such problems and Figure 29 to Figure 32 provides the simulation profiles on random samples of meshes.

4.1 Theta method using equation (1) on Ω= [0,1] × [0,1] where, 푓(푥, 푦) = Γ(푠 + 1)(푥푠 + 푦푠), (10) with initial and boundary conditions: 푢(푥, 0) = 0, 푢(0, 푦) = 0, 푢(푥, 1) = 푥푠, 푢(푦, 1) = 푦푠 the exact solution is 푢(푥, 푦) = (푥푦)푠, the fractional Poisson absolute error is identified by:

1 퐸푟푟표푟 = √∑푚−1 (푈 − 푢 )2, (11) (푚−1)2 푖,푗=1 푖,푗 푖,푗

Table 3 Solution of FPDES using meshes (Uniform Vs Random) Uniform Randomly generated grids Size Exact meshes First Second third 10x10 60.16820 57.79960 59.8660 59.0460 59.3210 20x20 60.16820 60.78560 60.0120 60.1120 59.3320 30x30 60.16820 56.74960 59.9870 60.7690 58.1240 40x40 60.16820 58.43960 58.9980 59.9760 60.4350 50x50 60.16820 56.79450 60.0010 59.6780 59.1240 60x60 60.16820 59.97960 60.1230 59.9920 59.5570 70x70 60.16820 58.79960 59.5460 59.2390 60.7790 80x80 60.16820 61.54960 59.8870 60.8970 59.1550 90x90 60.16820 57.79320 60.1240 58.6670 59.0010 100x100 60.16820 59.64320 59.3320 60.1120 60.1230

Table 4 Absolute Error for Different theta’s h=k theta= 0.9 theta=0.8 theta= 0.7 theta= 0.6 Ref.[26] s= 1:25 h=0.10 0.0072080 0.0067880 0.0062250 0.0016340 0.01380 h=0.05 0.0018840 0.0021190 0.0028370 0.0026280 0.00760 s= 1:50 h=0.10 0.0067530 0.0061830 0.0057760 0.0041340 0.00860 h=0.05 0.0019940 0.0018120 0.0015990 0.0025260 0.00450 s= 1:75 h=0.10 0.0062430 0.0052880 0.0057980 0.0037250 0.00630 h=0.05 0.0024240 0.0027370 0.0024890 0.0022320 0.00290

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Figure 6 Exact solution of 2D PDE’s Note that: solution of 2D Partial differential equations using uniform, random solutions (Figure 7- Figure 10) and Theta methods.

Figure 7 Local Solution profile on Uniform mesh of size 10x10

Figure 8 Local Solution profile on Uniform mesh of size 100x100

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Figure 9 Local Solution Profile on random mesh of size 10x10

Figure 10 Local Solution Profile on random mesh of size 100x100

5. CONCLUSION In this research, the numerical solution of second order PDE’s considered with the special treatment of randomly generated meshes. The numerical solutions of samples of random meshes were obtained and compared with the solutions over uniform meshes and parallelly theta method solve and analysed. The feasibility of using random meshes tested for each random sample in relation to their corresponding cell sizes and statistical parameters. The hypothesis was that the random meshes may improve the convergence of numerical solution. Therefore, the main objective was to test the practicability and feasibility of using the randomly generated finite difference meshes by statistical analysis of the random samples of meshes used to obtain the numerical solution.

FUTURE WORK In this research, extensive data analysis was done to test the feasibility of using random meshes but still there is much capacity to extend this research in different directions. For instance, the sensitivity of random mesh parameters can be analysed. The work can be extended in 3D Partial differential equations with different schemes over random meshes.

THE NOVEL IDEA In this paper, a novel method is proposed by Sanaullah Mastoi.

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