# Numerical Solution of the Multiterm Time-Fractional Model for Heat Conductivity by Local Meshless Technique

Authors: Bander N. Almutairi; Ahmed E. Abouelregal; Bandar Bin-Mohsin; M. D. Alsulami; Phatiphat Thounthong
Journal: Complexity
 (2021)

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## Abstract

Fractional partial differential equation models are frequently used to several physical phenomena. Despite the ability to express many complex phenomena in different disciplines, researchers have found that multiterm time-fractional PDEs improve the modeling accuracy for describing diffusion processes in contrast to the results of a single term. Nowadays, it attracts the attention of the active researchers. The aim of this work is concerned with the approximate numerical solutions of the three-term time-fractional Sobolev model equation using computationally attractive and reliable technique, known as a local meshless method. Because of the meshless character and the simple application in higher dimensions, there is a growing interest in meshless techniques. To assess the reliability and accuracy of the proposed method, three test problems and two types of irregular domains are taken into account.


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## Body

## 1. Introduction

In recent years, fractional partial differential equations (FPDEs) have drawn the consideration of numerous researchers to their applications in various fields of science and technology. Partial derivatives provide a flexible model and an extraordinary tool for description of capturing the history of the variable and genetic characteristics of various dynamic systems. Extensive research has been carried out in the advancement of numerical and analytical solutions of linear and nonlinear FPDEs [1–6]. However, several researchers have not succeeded in deriving and modeling many complex phenomena utilizing linear or nonlinear PDEs with integer order [7]. Subsequently, the fractional is taken as account and is a good solution to this problem [8]. In the current work, three-term time-fractional Sobolev equation is considered which can be expressed as(1)∂β1Vy,z,t∂tβ1+∂β2Vy,z,t∂tβ2+∂β3Vy,z,t∂tβ3−∂∇2Vy,z,t∂t−β∇2Vy,z,t+γ∇Vy,z,t∇Vz¯,t+δVy,z,t=Fz¯,t,y,z∈Ω,0<β3≤β2≤β1≤1,t>0.With the conditions,(2)Vy,z,0=V0y,z,Vy,z,t=g1y,z,t,y,z∈∂Ω,where ∇2 is the Laplacian and ∇ denotes gradient operators, and β, γ, and δ the are known constants, whereas ∂β1/∂tβ1, ∂β2/∂tβ2, and ∂β3/∂tβ3 represent the Caputo derivative operator of order 0<β3≤β2≤β1≤1 for the function Vy,z,t.In recent literature, various meshless methods have been utilized for the numerical solution of various PDE models almost in every discipline of science and engineering. In particular, the RBF-based meshless methods are the mainstream of these methods. The meshless nature is one of the main reasons behind the developing interest for such approaches. The meshless methods significantly reduce the complexity of dimensionality utilizing traditional methods such as the finite element and finite difference methods. Compared to mesh-based methods, these methods do not require mesh in the domain. The meshless methods have the ability to compute the solution in regular and irregular domain utilizing scattered or uniform nodes, which increases the priority and the advantages of meshless methods. As these facts show, these methods are really workable and useful numerical methods that can be applied to real-world challenging problems [9–17].The RBF-based meshless methods have also some deficiencies like other numerical methods, in which the most important one is the dense ill-conditioned matrices and the selection of the optimal value of the shape parameter. To avoid these drawbacks, local meshless methods are the best alternatives, suggested by the researchers which are considered to be accurate and stable for the solution of diverse integer and fractional-order PDE models [18, 19]. The local meshless methods are less sensitive to the change in shape parameters than the global version, and it produces well-conditioned sparse matrices. Furthermore, local version of meshless methods is considered to be more effective and efficient than global ones. In recent years, the abilities of various sorts of local meshless methods in different applications have been explored [20–22].In the current research, we have implemented the local meshless method to approximate the numerical solution of three-term time-fractional model equation (1). For this purpose, multiquadric (MQ) radial basis functions (RBFs) are used. Furthermore, two types of irregular domains are also taken in numerical examples.


## 2. Methodology of the Local Meshless Method

According to the local meshless method, to approximate the derivatives ofVz¯,t at the centers z¯h by the neighborhood of z¯h, z¯h1,z¯h2,z¯h3,…,z¯hnh⊂z¯1,z¯2,…,z¯Nn,nh≪Nn, where h=1,2,…,Nn, we have used z¯=y and z¯=y,z for one-dimensional and two-dimensional cases, respectively.Now, considering the following case for one-dimensional,(3)Vmyh≈∑k=1nhλkmVyhk,h=1,2,…,N.Substituting the multiquadric RBFψy−yp=1+cyhk−yp2 in (3),(4)ψmyh−yp=∑k=1nhλhkmψyhk−yp,p=h1,h2,…,hnh.Equation (4) in matrix form is(5)ψh1myhψh2myh⋮ψhnhmyh︸ψnhm=ψh1yh1ψh2yh1⋯ψhnhyh1ψh1yh2ψh2yh2⋯ψhnhyh2⋮⋮⋱⋮ψh1yhnhψh2yhnh⋯ψhnhyhnh︸Anhλh1mλh2m⋮λhnhm︸λnhm,where(6)ψpyk=ψyk−yp,p=h1,h2,…,hnh,for each k=i1,h2,…,hnh. Equation (5) in simple form is(7)ψnhm=Anhλnhm.From (7), we obtain(8)λnhm=Anh−1ψnhm.(3) and (8) implies(9)Vmyh=λnhmTVnh,where(10)Vnh=Vyh1,Vyh2,…,VyhnhT.The derivatives ofVy,z,t w. r. t. y and z can be found as(11)Vymyh,zh≈∑k=1nhγkmVyhk,zhk,h=1,2,…,N2,Vymyh,zh≈∑k=1nhηkmVyhk,zhk,h=1,2,…,N2.Forγkm and ηkmk=1,2,…,nh, we continue as(12)γnhm=Anh−1Φnhm,ηnhm=Anh−1Φnhm.The time derivative∂β1Vz¯,t/∂tβ1 is discretized utilizing Caputo derivative [23], where β1∈0,1 as(13)∂β1Vz¯,t∂tβ1=1Γ1−β1∫0t∂Vz¯,ϑ∂ϑt−ϑ−β1dϑ,0<β1<1,∂Vz¯,t∂t,β1=1.Letτ be the time step size, and for the interval 0,t, consider tq=qτ, q=0,1,2,…,Q. We complete the time-fractional derivative term as(14)∂β1Vz¯,tq+1∂tβ1=1Γ1−β1∫0tq+1∂Vz¯,ϑ∂ϑtq+1−ϑ−β1dϑ,=1Γ1−β1∑r=0q∫rτr+1τ∂Vz¯,ϑ∂ϑtr+1−ϑ−β1dϑ,≈1Γ1−β1∑r=0q∫rτr+1τ∂Vz¯,ϑr∂ϑtr+1−ϑ−β1dϑ.The term∂Vz¯,ϑr/∂ϑ is approximated as follows:(15)∂Vz¯,ϑr∂ϑ=Vz¯,ϑr+1−Vz¯,ϑrϑ+Oτ.Then,(16)∂β1Vz¯,tq+1≈1/Γ1−β1∑r=0qVz¯,tr+1−Vz¯,tr/τ∫rτr+1τtr+1−ϑ−β1dϑ∂tβ1,=1Γ1−β1∑r=0qVz¯,tq+1−r−Vz¯,tq−rτ∫rτr+1τtr+1−ϑ−β1dϑ,=τ−β1Γ2−β1Vq+1−Vq+τ−β1Γ2−β1∑r=1qVq+1−r−Vq−rr+11−β1−r1−β1,q≥1,τ−β1Γ2−β1V1−V0,q=0.Lettinga0=τ−β1/Γ2−β1 and br=r+11−β1−r1−β1, r=0,1,…,q, we have(17)∂β1Vz¯,tq+1∂tβ1≈a0Vq+1−Vq+a0∑r=1qbrVq+1−r−Vq−r,q≥1,a0V1−V0,q=0.The fractional derivative of orderβ2 and β3 can be found as above.


## 3. Numerical Experiments

This section examines the accuracy and applicability of the proposed method for the three-term time-fractional model (1). In the test problems, we have considered regular and irregular domains. This computation is considered to be regular and scattered nodes with regular and irregular domains. In this article, we have used the Crank–Nicholson scheme and multiquadric (MQ) RBF with shape parameter valuec=10. Unless specifically stated, the spatial domain 0,4 and time step size τ=0.002 are used. Accuracy is measured as follows:(18)Labsolute=V^−V,Max−error=maxLabsolute,RMS=∑h=1NnV^h−Vh2N,where V^ is the exact solution, and V is the approximate solution.Problem 1.
Consider the model equation:(19)∂β1Vy,z,t∂tβ1+∂β2Vy,z,t∂tβ2+∂β3Vy,z,t∂tβ3−∂∇2Vy,z,t∂t−∇2Vy,z,t=Fy,z,t,0<β3≤β2≤β1≤1,t>0.
Having the exact solution,(20)Vy,z,t=e−tsinπysinπz,y,z∈Ω.
The proposed meshless method is implemented for generating the required numerical results for Problem1, which are given in Table 1. Different values of a number of nodes N, fractional order β1=β2=β3, and final time t=1 are used, whereas the error norms stand for max−error and RMS. These results revealed the fact that the recommended meshless method is capable of better results. Showing the accurate and efficient of the method, the results are compared with the exact solution for β1=β2=β3=0.1, β1=β2=β3=0.3, β1=β2=β3=0.5, t=1, t=2, and for various values of time step size τ. These results are computed using N=82 and are given in Table 2. One can observe from this table that only in few iterations, the suggested meshless method produced better results, and as the number of time iteration increases, the accuracy increase and the error norm reached up to max−error≈10−10. As the condition number, stability, and accuracy of the RBF-based meshless methods heavily depend on the value of shape parameter c, a little change in shape parameter value causes instability and the results get diverge. But the suggested local meshless method is tested for Problem 1 in terms of condition number, stability, and accuracy as shown in Figure 1 for N=102, β1=β2=β3=0.5, and t=1. This figure revealed that the suggested meshless method is stable, accurate, and given ideal low condition number ≈1 for a long range of c up to 2000. Figure 2 shows the absolute error using β1=β2=β3=0.1 and β1=β2=β3=0.8 for N=102 and t=1. Better accuracy of the recommended algorithm can be seen in this figure.Table 1
Problem1, approximate results for t=1.
β1=β2=β3=0.2β1=β2=β3=0.5β1=β2=β3=0.8NMax−errorRMSMax−errorRMSMax−errorRMS828.5869e − 083.9708e − 083.8841e − 071.2258e − 074.9568e − 061.7762e − 061028.2120e − 083.9848e − 084.8466e − 071.4140e − 075.8961e − 061.9704e − 061228.8415e − 083.9397e − 085.2865e − 071.6405e − 076.1379e − 062.1931e − 06Table 2
Problem1, approximate results using N=82 and β=β1=β2=β3.
Max−errorτt=1t=2β=0.1β=0.3β=0.5β=0.1β=0.3β=0.50.21.1044e − 039.9648e − 047.9676e − 047.9063e − 047.1256e − 045.7016e − 040.021.0790e − 058.0577e − 063.9095e − 067.7071e − 065.6947e − 063.1952e − 060.0021.0500e − 074.3655e − 083.8841e − 077.4936e − 083.0435e − 082.9006e − 07Figure 1
Problem1. (a) c and error norms, (b) c and condition number.
(a)(b)Figure 2
Problem1. Absolute error for β=0.1 (a) and β=0.8 (b), where β=β1=β2=β3.
(a)(b)Problem 2.
Consider the model equation:(21)∂β1Vy,z,t∂tβ1+∂β2Vy,z,t∂tβ2+∂β3Vy,z,t∂tβ3−∂∇2Vy,z,t∂t−∇2Vy,z,t=Fy,z,t,0<β3≤β2≤β1≤1,t>0.
Having the exact solution,(22)Vy,z,t=ey−z−tsinπysinπz,y,z∈Ω.
In Table3, we have implemented the suggested algorithm for generating the numerical results for Problem 2 for N=82, N=102, N=122, β1=β2=β3=0.2, β1=β2=β3=0.4, β1=β2=β3=0.4, and t=1, t=2. The results are assessed in term of max−error and RMS. Accurate results have been obtained in this problem as well. Showing the applicability and efficacy of the propose method, the results are compared with the exact solution for various values of β1=β2=β3t and τ using N=82. These results are given in Table 4. One can observe from this table that only in few iterations, the suggested meshless method produced better results, and as the number of time iteration increases, the accuracy increase and the error norm reached up to RMS≈10−9.
Just like the previous problem, the suggested method has been tested for Problem2 in terms of condition number, stability, and accuracy as shown in Figure 3 for N=102, β=0.5, and t=1. It can easily be seen from the figure that the suggest meshless method is stable, accurate, and given ideal low condition number ≈1 for a long range of c up to 2000, whereas in Figure 4, we have shown a comparison of exact and approximate solutions for various values of time t and brilliant match of both the solutions can be found in this figure.
One of the principle advantages of the meshless techniques over mesh-based techniques is the implementation in the irregular domain with ease. In this article, two types of challenging irregular domains are taken into account, which are displayed in Figure5. In Table 5, we have shown the numerical results obtained by the suggested meshless method corresponding to the irregular domains for Problem 1 and Problem 2. We have considered the various value of β ’s, and the results are shown in form of max−error and RMS. It is observed from the table that better accuracy has been achieved in both domains.Table 3
Problem2, approximate results for t=1.
Nβ1=β2=β3=0.2β1=β2=β3=0.4β1=β2=β3=0.6Max−errorRMSMax−errorRMSMax−errorRMS821.2325e − 062.0404e − 072.3271e − 063.3074e − 071.9346e − 052.5361e − 061021.5895e − 062.1784e − 073.2329e − 064.5136e − 072.6251e − 053.2570e − 061221.4030e − 062.1123e − 074.2754e − 065.8259e − 073.1278e − 053.9590e − 06Table 4
Problem2, approximate results for N=82.
RMSτt=1t=2β=0.2β=0.4β=0.6β=0.2β=0.4β=0.60.22.6139e − 032.2018e − 031.5260e − 031.8624e − 031.5697e − 031.0945e − 030.022.3792e − 051.2980e − 054.3173e − 051.6857e − 059.2597e − 063.3106e − 050.0022.0404e − 073.3074e − 072.5361e − 061.4425e − 072.4742e − 071.9042e − 060.00021.5812e − 091.1891e − 081.1014e − 071.1152e − 098.7439e − 098.2401e − 08Figure 3
Problem2. (a) c and error norms. (b) c and condition number.
(a)(b)Figure 4
Problem2, approximate and exact solution for indicated time.Figure 5
Computational domain 1 (a) and domain 2 (b).
(a)(b)Table 5
Approximate results corresponding to the irregular domains.
Domainβ=0.2β=0.5β=0.8Max−errorRMSMax−errorRMSMax−errorRMSProblem1Domain 12.8188e − 077.0294e − 083.2180e − 068.2127e − 073.3601e − 059.0154e − 06Domain 21.9344e − 063.4442e − 072.0244e − 053.8603e − 061.1285e − 042.6506e − 05Problem2Domain 19.0788e − 061.1575e − 069.5402e − 051.0099e − 051.0994e − 031.1639e − 04Domain 23.3707e − 066.0517e − 073.2748e − 056.3853e − 061.2824e − 043.2931e − 05Problem 3.
Consider the model equation:(23)∂β1Vy,z,t∂tβ1+∂β2Vy,z,t∂tβ2+∂β3Vy,z,t∂tβ3−∂∇2Vy,z,t∂t−∇2Vy,z,t+∇Vy,z,t∇Vz¯,t+π2Vy,z,t=Fy,z,t,y,z∈Ω,0<β3≤β2≤β1≤1,t>0.
Having the exact solution,(24)Vy,z,t=etsinπysinπz,y,z∈Ω.
In Figure6, we have visualized the behavior of the exact and approximate solutions for the Problem 3 using N=202, β1=β2=β3=0.5, and t=0.1, which show that the approximate solution is very compatible with the exact solution. In Figure 7, the absolute error is displayed for Problem 3.Figure 6
Problem3. (a) Exact solution. (b) Numerical solution.
(a)(b)Figure 7
Problem3, absolute error at (a) t=0.01 and (b) t=0.05.
(a)(b)


## 4. Conclusion

In this study, our principle focused on the applicability and performance of the RBF-based local meshless method to approximate the numerical solution of three-term time-fractional Sobolev equations. The computed results show that the proposed technique can take care of these sorts of problems amazingly and accurately. The local procedure leads to a sparse system of linear equations, and the solution is approximated with good accuracy. Three test problems are taken into account to test the effectiveness and accuracy of the proposed meshless method utilizing rectangular and two irregular domains. The numerical results demonstrate the high accuracy and effectiveness of the method. Given the current research, the proposed technique is a surprisingly powerful and successful tool for solving numerical problems of multiterm time-fractional PDEs found in various fields of science and technology.



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