# A Hybrid Interpolation Method for Fractional PDEs and Its Applications to Fractional Diffusion and Buckmaster Equations

Authors: Ihtisham Ul Haq; Nigar Ali; Shabir Ahmad; Tayyaba Akram
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study presents a novel numerical method to solve PDEs with the fractional Caputo operator. In this method, we apply the Newton interpolation numerical scheme in Laplace space, and then, the solution is returned to real space through the inverse Laplace transform. The Newton polynomial provides good results as compared to the Lagrangian polynomial, which is used to construct the Adams–Bashforth method. This procedure is used to solve fractional Buckmaster and diffusion equations. Finally, a few numerical simulations are presented, ensuring that this strategy is highly stable and quickly converges to an exact solution.


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## Body

## 1. Introduction

Recently, partial differential equations (PDEs) have been successfully used to handle many real problems in engineering and mathematical physics such as viscoelastic materials, seismic analysis, and viscous damping [1–3]. Over the past few decades, fractional calculus has been used for better analysis of mathematical models [4, 5]. Fractional operators have been implemented to study nonlocal behaviour of the physical model. Khan et al. investigated hidden attractors in four-dimensional dynamical systems with power law kernel [6]. Khan et al. analyzed the imperfect testing infection disease model using the fractional operator [7]. Some other applications of fractional calculus are listed in [8, 9]. Also, fractional PDEs have attracted the scientists due to their wide range of applications in various areas of science. For instance, Saifullah et al. investigated the Klein–Gordon equation with power law kernel [10]. Gulalai et al. analyzed nonlinear modified KdV equation under Atangana–Baleanu–Caputo derivative [11]. Fractional PDEs have memory terms; they are not the same as classical PDEs, and evaluating FPDEs numerically is much difficult than solving PDEs. So, it is essential to establish effective numerical techniques to deal with PFDEs. There are several analytical and numerical methods for solving fractional PDEs. Akram et al. studied applied extended cubic B-spline collocation scheme for time fractional subdiffusion equation, cable equation, and Klein–Gordan equation [12–14]. Ahmad et al. used Yang transfrom coupled with the homotopy perturbation method (HPM) to study fractional KdV and Fornberg–Whitham equations [15]. The Legendre wavelets method has been utilized to investigate parabolic fractional PDEs [16]. A biorthogonal Hermite cubic spline Galerkin method has been used to solve fractional Riccati equation [17]. Iqbal et al. applied the Elzaki transformation and the new iteration technique to study fractional-order Kaup–Kupershmidt equation [18]. Shah et al. utilized the Haar wavelet collocation method to analyze the fractional-order COVID-19 model [19]. Many other approaches have been used for computing solution of fractional PDEs [20, 21].The Adams–Bashforth method has been recognized as a powerful numerical scheme for solving nonlinear partial differential equations with integer and noninteger derivatives. Gnitchogna and Atangana proposed a new numerical scheme which can be seen as an extension of the Adams–Bashforth one for partial differential equations using Lagrangian interpolation [22]. There is a need to expand Newton interpolation numerical schemes to PFDEs because of their accuracy and efficiency in integer-order and fractional derivatives [23]. To handle PFDE with Caputo derivative, we develop a numerical scheme that combines the Newton interpolation numerical scheme and Laplace transform. By removing one variable and converting a PFDE to an FDE using the Laplace transform, the newly generated FDE can be easily solved in Laplace space, and the result may then be returned to real space using the inverse transform. Finally, some examples are provided to assess the efficacy of the developed method.The study is organized as follows: Section2 provides the basic concepts of fractional calculus. Section 3 deals with numerical approximation of fractional PDEs using Newton polynomial. Sections 4 and 5 are the applications of the suggested scheme. Section 6 gives the numerical simulations of the proposed results. Last, the conclusion is provided in Section 7.


## 2. Basic Concepts of Caputo Operator

Here, we recall important definitions of Laplace transform and fractional derivatives [24].Definition 1.
The Caputo-type fractional derivative is given as(1)c0Dtμ1Yt=1Γp−μ1∫0tt−ξp−μ1−1ddξYξdξ,where p−1<μ1<p∈N.Definition 2.
The Laplace transform (LT) of the functionYt is(2)Ys=ℒYt=∫0∞e−stYtdt.Definition 3.
The LT ofc/0Dtμ1Yt is(3)ℒc0Dtμ1Yts=Sμ1Ys−∑p=0p−1Sk−μ1−1Yp0,where p−1<μ1<p∈N.


## 3. Numerical Approximations

Consider a general fractional nonlinear PDE as(4)∂μ1Yη,t∂tμ1=LYη,t+ℕYη,t,where ℕ is a nonlinear operator and L is a linear operator. Applying LT to (4), we get(5)ℒ∂μ1Yη,t∂tμ1=ℒLYη,t+ℒℕYη,t.Then,(6)c0Dtμ1Ys,t=ℒLYη,t+ℒℕYη,t.LetYs,t=Yt and Gt,Yt=ℒLYη,t+ℒℕYη,t. Then, (6) becomes(7)c0Dtμ1Yt=Gt,Yt,0<t≤T,0<μ1≤1,Y0=Y0.Using the Caputo fractional integral operator, we get(8)Yt=Y0+1Γμ1∫0tt−ξμ1−1Gξ,Yξdξ.Fortp+1, one gets(9)Yp+1=Y0+1Γμ1∫0tp+1tp+1−ξμ1−1Gξ,Yξdξ.Next, we approximate the above equation as(10)Yp+1=Y0+1Γμ1∑m=0p∫tmtm+1tp+1−ξμ1−1Gξ,Yξdξ,where t0=0. So, the kernel Gξ,Yξ might be approximated with in tp,tp+1 via Newton’s polynomial. So,(11)Pmξ=Gtm,Ym+Gtm+1,Ym+1−Gtm,YmΔtξ−tmifm=0,Gtm+1,Ym+1+Gtm+1,Ym+1−Gtm,YmΔtξ−tm+1ifm∈1,2,…,p,+Gtm+1,Ym+1−2Gtm,Ym+Gtm−1,Ym−12Δt2ξ−tmξ−tm+1.Substituting equation (11) in (10), we get(12)Yp+1=Y0+1Γμ1∫0t1G0+G1−G1Δtξtp+1−ξα−1dξ+1Γμ1∑m=1p∫tmt1+mGm+−Gm+G1+mΔtξ−t1+m+Gm+1−2Gm+G−1+m2Δt2ξ−tmξ−t1+mtp+1−ξμ1−1dξ.Using the linearity property of integral, we have(13)Yp+1=Y0+1Γμ1G0∫0t1tp+1−ξμ1−1dξ+1Γμ1G1−G0Δt∫0t1ξtp+1−ξμ1−1dξ+1Γμ1∑m=1pGm+1∫tmtm+1tp+1−ξα−1dξ+1Γμ1∑m=1pGm+1−GmΔt∫tmtm+1ξ−tm+1tp+1−ξμ1−1dξ+1μ1∑m=1pGm+1−2Gm+Gm−12Δt2∫tmtm+1ξ−tmξ−tm+1tp+1−ξμ1−1dξ.Now, we evaluate the integrals in (13) as(14)∫0t1ξtp+1−ξμ1−1dξ=Δtμ1+1μ1μ1+1−μ1+p+1μ1+p+1μ1+1−pμ1+1,(15)∫tmtm+1tp+1−ξμ1−1dξ=Δtμ1μ1p+1−mμ1−−m+pμ1,(16)∫tmtm+1ξ−tm+1tp+1−ξμ1−1dξ=Δtμ1μ1μ1+1p+μ1−mp+1−mμ1−−m+pμ1+1,(17)∫tmtm+1ξ−tmξ−tm+1tp+1−ξdξ=Δtμ1+2μ1μ1+1μ1+2p+1−mμ12−m+p2−μ1−m+1+p+2−m+p−−m+pμ12−m+p2−μ1−m+p+2p−m.By substituting these integrals values into (13), we get(18)Yp+1=Y0+1Γμ1G0Δtμ1μ1p+1μ1−pμ1+1Γμ1G1−G0ΔtΔtμ1+1μ1μ1+1p+1μ1+1−pμ1+1−μ1+1pμ1+1Γμ1∑m=1pGm+1−GmΔt∫tmtm+1ξ−tm+1tp+1−ξμ1−1dξ+1Γμ1∑m=1pGm+1Δtμ1μ1−m+p+1μ1−−m+pμ1+1Γμ1∑m=1pGm+1−2Gm+Gm−12Δt2Δtμ1+2μ1μ1+1μ1+2p+1−mμ12−m+p2−μ1p+1−m+2−m+p−−m+pμ12−m+p2−μ1−m+p+2−m+p.Rearranging the terms and simplifying, we have(19)Yp+1=Y0+Δtμ1Γμ1+1G0p+1μ1−pμ1Δtμ1−1Γμ1+2G1−G0p+1μ1+1−pμ1+1−μ1−1pμ1+Δtμ1Γμ1+1∑m=1pGm+1−m+p+1μ1−−m+pμ1+Δtμ1−1Γμ1+2∑m=1pGm+1−Gmp+μ1−mp+1−mμ1−−m+pμ1+1+Δtμ12Γμ1+3∑m=1pGm+1−2Gm+Gm−1p+1−mμ12−m+p2−μ1p+1−m+2−m+p−−m+pμ12−m+p2+2−m+p−μ1−m+p.The inverse Laplace transform is used to get the solution as(20)Yη,tp+1=ℒ−1Y0+Δtμ1Γμ1+1G0p+1μ1−pμ1Δtμ1−1Γμ1+2G1−G0p+1μ1+1−pμ1+1−μ1−1pμ1+Δtμ1Γμ1+1∑m=1pGm+1p+1−mμ1−−m+pμ1+Δtμ1−1Γμ1+2∑m=1pGm+1−Gmp+μ1−mp+1−mμ1−−m+pμ1+1+Δtμ12Γμ1+3∑m=1pGm+1−2Gm+Gm−1p+1−mμ12−m+p2−μ1p+1−m+2−m+p−−m+pμ12−m+p2−μ1−m+p+2−m+p.Furthermore, we can discretize equation (16) as follows:(21)Yηm,tp+1=Y0+ℒ−1Δtμ1Γμ1+1G0p+1μ1−pμ1Δtμ1−1Γμ1+2G1−G0p+1μ1+1−pμ1+1−μ1−1pμ1+Δtμ1Γμ1+1Gm+1pp+1−mμ1−−m+pμ1+Δtμ1−1Γμ1+2Gm+1p−Gmpp+μ1−mp+1−mμ1−−m+pμ1+1+Δtμ12Γμ1+3Gm+1p−2Gmp+Gm−1pp+1−mμ12−m+p2−μ1p+1−m+2−m+p−−m+pμ12−m+p2−μ1−m+p+2−m+p,where gηm,tp=gmp. Equation (17) gives the numerical approximation of the fractional PDEs using Newton polynomial.In the following sections, this novel method will be applied to solve a fraction diffusion problem and a fractional Buckmaster equation.


## 4. Applications to Fractional Diffusion Equation

Diffusion equation is parabolic PDE having many applications in real world phenomena. In physics, it represents the macroscopic behaviour of multiple microparticles in Brownian motion, which is caused by the particles’ random motions and collisions. Here, we evaluate the accuracy and effectiveness of the newly proposed numerical method by applying it to a fractional diffusion model. Consider(22)∂μ1gz,t∂tμ1=C∂2gz,t∂z2.Using Laplace transform, we get(23)c0Dtμ1gs,t=C−g0,t−sg0,t+s2gs,t.Letgs,t=gt and Gt,gt=s2gs,t−sg0,t−g0,t.Equation (19) can be rewritten as(24)c0Dtμ1gt=CGt,gt.We already established that the solution to equation (22) in the Laplace space is given in equation (15). So,(25)gtp+1=Cgt0+CΔtμ1μ1+1G0tp+1μ1−pμ1CΔtμ1−1μ1+2G1t−G0tp+1μ1+1−pμ1+1−μ1−1pμ1+CΔtμ1μ1+1∑m=1pGm+1tp+1−mμ1−−m+pμ1+Δtμ1−1μ1+2∑m=1pCGm+1t−Gmt−−m+pμ1+1+p+μ1−mp+1−mμ1+Δtμ12μ1+3∑m=1pCGm+1t+Gm−1t−2Gmt−m+p+1μ12−m+p2−μ1p+1−m+2−m+p−−m+pμ12−m+p2−μ1−m+p+2−m+p.Let us define the expression to simplify the formulas as(26)gzm,tp+1=Cgt0+CΔtμ1Γμ1+1δ1pG0+CΔtμ1−1Γμ1+2G1−G0δ2p+pΔtμ1Γμ1+1Gm+1pδ3p+CΔtμ1−1Γμ1+2CGm+1p−Gmpδ4p+Δtμ12Γμ1+3CGm+1p−2Gmp+Gm−1pδ5p,where δ1p=p+1μ1−pμ1,δ2p=p+1μ1+1−pμ1+1−μ1−1pμ1,δ3p=p+1−mμ1−−m+pμ1, δ4p=μ1−m+p−m+p+1μ1−−m+pμ1+1,δ5p=2−m+p2−μ1p+1−m+2−m+p,−−m+pμ12−m+p+2−m+p2−μ1−m+p.The inverse Laplace transform is used to get the solution as(27)gtp+1=Cgt0+CΔtμ1Γμ1+1δ1p∂2gz,t0∂z2+δ2pCΔtμ1−1Γμ1+2∂2gz,t1∂z2−∂2gz,t0∂z2+δ3pCΔtμ1Γμ1+1∂2gz,tm+1∂z2+δ4pCΔtμ1−1Γμ1+2∂2gz,tm+1∂z2−∂2gz,tm∂z2+δ5pCΔtμ12Γμ1+3∂2gz,tm+1∂z2−2∂2gz,tm∂z2+∂2gz,tm−1∂z2.Discretizing in the space variable, we get(28)gtp+1=Cgt0+CΔtμ1Γμ1+1δ1pgzm,t0+gzm+1,t0−gzm,t0Δt+δ2pCΔtμ1−1Γμ1+2gzm+1,t1+gzm+1,t1−gzm,t1Δt+gzm+1,t1−2gzm,t1+gzm−1,t12Δt2−gzm,t0+gzm+1,t0−gzm,t0Δt+δ3pCΔtμ1Γμ1+1gzm+1,t1gzm+1,tp+1−gzm,tp+1Δt+gzm+1,tp+1−2gzm,tp+1+gzm−1,tp+12Δt2+δ4pCΔtμ1−1Γμ1+2gzm+1,tp+1+gzm+1,tp+1−gzm,tp+1Δt+gzm+1,tp+1−2gzm,tp+1+gzm−1,tp+12Δt2−gzm+1,tp+gzm+1,tp−gzm,tpΔt+gzm+1,tp−2gzm,tp+gzm−1,tp2Δt2+Cδ5pΔtμ12Γμ1+3gzm+1,tp+1+gzm+1,tp+1+gzm,tp+1Δt+gzm+1,tp+1−2gzm,tp+1+gzm−1,tp+12Δt2−2gzm+1,tp+gzm+1,tp−gzm,tpΔt+gzm+1,tp−2gzm,tp+gzm−1,tp2Δt2+gzm+1,tp−1+gzm+1,tp−1−gzm,tp−1Δt+gzm+1,tp−1−2gzm,tp−1+gzm−1,tp−12Δt2.Lettinggzm,tp=gmp and Δt=h, equation (24) becomes(29)gtp+1=Cgt0+δ1pChμ1Γμ1+1gm0+gm+11−gm0h+δ2pChμ1−1Γμ1+2gm+11+gm+11−gm1h+gm+11−2gm1+gm−112h2−gm0+gm+11−gm0h+δ3pCΔtμ1Γμ1+1gm+11+gm+1p+1−gm+1ph+gm+1p+1−2gmp+1+gm−1p+12h2+δ4pChμ1−1Γμ1+2gm+1p+1+gm+1p+1−gm+1ph+gm+1p+1−2gmp+1+gm−1p+12h2−gm+1p+gm+1p−gmph+gm+1p−2gmp+gm−1p2h2+Cδ5phμ12Γμ1+3gm+1p+1+gm+1p+1−gmp+1h+gm+1p+1−2gmp+1+gm−1p+12h2−2gm+1p+gm+1p−gmph+gm+1p−2gmp+gm−1p2h2+gm+1p−1+gm+1p−1−gmp−1h+gm+1p−1−2gmp−1+gm+1p−12h2.Thus, equation (25) is the required numerical results for fractional diffusion equation using Newton polynomial.


## 5. Application to the Fractional Buckmaster Equation

The Buckmaster equation is a nonlinear PDE that was developed in 1977 by John D. Buckmaster. The surface of a thin sheet of viscous liquid is represented by this equation. In this section, we provide another application of the proposed scheme to show the its validity. We consider the fractional Buckmaster equation as(30)c0Dtμ1yz,t=∂y4∂z2+∂y3∂z.Taking the Laplace transform of (29), we get(31)ℒc0Dtμ1yz,t=ℒ∂y4∂z2+∂y3∂z,(32)c0Dtμ1ys,t=ℒ∂y4∂z2+∂y3∂z.Letys,t=yt and ℒ∂y4/∂z2+∂y3/∂z=ℒy4zz+y3z=Yy,t. Now, consider(33)c0Dtμ1yt=Yy,t.We already established that the solution to (28) in Laplace space is represented by (15). So,(34)yp+1=y0+Δtμ1Γμ1+1Y0k+1μ1−kμ1Δtμ1−1Γμ1+2Y1−Y0p+1μ1+1−pμ1+1−μ1−1pα+Δtμ1Γμ1+1Ym+1p−m+p+1μ1−−m+pμ1+Δtμ1−1Γμ1+2Ym+1p−Ymp−−m+pμ1+1+p+μ1−mp+1−mμ1+Δtα2Γμ1+3Ym+1p−2Ymp+Ym−1pp+1−mμ12−m+p2−μ1p+1−m2−m+p−−m+pμ12−m+p2−μ1−m+p+2−m+p.Taking the inverse Laplace transform, we get(35)gz,tp+1=gt0+Δtμ1Γμ1+1δ1pyzz4+yz3t0+δ2kΔtμ1−1Γμ1+2yzz4+yz3t1−yzz4+yz3t0+δ3pΔtμ1Γμ1+1yzz4+yz3tm+1+δ4pΔtμ1−1Γμ1+2yzz4+yz3tm+1−yzz4+yz3tm+δ5pΔtμ12Γμ1+3yzz4+yz3tm+1−2yzz4+yz3tm+yzz4+yz3tm−1,where δ1p=p+1μ1−pμ1,δ2p=p+1μ1+1−pμ1+1−μ1−1pμ1, δ3p=−m+p+1μ1−−m+pμ1,δ4p=p+μ1−mp+1−mμ1−−m+pμ1+1,δ5p=2−m+p+2−m+p2−μ1p+1−m,−−m+pμ12−m+p2+2−m+p−μ1−m+p.Discretizing in the space variable and substitutingΔt=h, we get(36)gzm,tp+1=gt0+δ1phμ1Γμ1+1y4zm,t0+y4zm+1,t0−y4zm,t0h+y3zm,t0+y3zm+1,t0−y3zm,t0h+δ2phα−1Γα+2y4zm+1,t1+y4zm+1,t1−y4zm,t1h+y4zm+1,t1−2y4zm,t1+y4zm−1,t12h2+y3zm+1,t1+y3zm+1,t1−y3zm,t1h+y3zm+1,t1−2y3zm,t1+y3zm−1,t12h2y4zm,t0+y4zm+1,t0−y4zm,t0h+y3zm,t0+y3zm+1,t0−y3zm,t0h+δ3phμ1Γμ1+1y4z1+m,t1+my4z1+m,t1+m−y4zm,tm+1h+y4z1+m,t1+m−2y4zm,tm+1+y4zm−1,tm+12h2+y3z1+m,t1+m+y3z1+m,t1+m−y3zm,tm+1h+y3z1+m,t1+m−2y3zm,tm+1+y3zm−1,tm+12h2+δ4phμ1−1Γμ1+2y4z1+m,t1+m+y4z1+m,t1+m−y4zm,tm+1h+y4z1+m,t1+m−2y4zm,tm+1+y4zm−1,tm+12h2y3z1+m,t1+m+y3z1+m,t1+m−y3zm,tm+1h+y3z1+m,t1+m−2y3zm,tm+1+y3zm−1,tm+12h2−y4zm+1,tm+y4zm+1,tm−y4zm,tmh+y4zm+1,tm−2y4zm,tm+y4zm−1,tm2h2+y3zm+1,tm+y3zm+1,tm−y3zm,tmh+y3zm+1,tm−2y3zm,tm+y3zm−1,tm2h2+δ5phμ12Γα+3y4z1+m,t1+m+y4z1+m,t1+m−y4zm,tm+1Δt+y4z1+m,t1+m−2y4zm,tm+1+y4zm−1,tm+12h2y3z1+m,t1+m+y3z1+m,t1+m−y3zm,tm+1h+y3z1+m,t1+m−2y3zm,tm+1+y3zm−1,tm+12h2−2y4zm+1,tm+y4zm+1,tm−y4zm,tmh+y4zm+1,tm−2y4zm,tm+y4zm−1,tm2h2y3zm+1,tm+y3zm+1,tm−y3zm,tmh+y3zm+1,tm−2y3zm,tm+y3zm−1,tm2h2y4zm+1,tm−1+y4zm+1,tm−1−y4zm,tm−1h+y4zm+1,tm−1−2y4zm,tm−1+y4zm−1,tm−12h2y3zm+1,tm−1+y3zm+1,tm−1−y3zm,tm−1h+y3zm+1,tm−1−2y3zm,tm−1+y3zm−1,tm−12h2.Thus, equation (31) gives the numerical solution of the proposed fractional Buckmaster equation.


## 6. Numerical Simulations

In this section, we demonstrate numerical solutions to diffusion and Buckmaster equations for various fractional orders. Figures1–6 show the simulation of the numerical solution of the fractional diffusion equation for fractional order μ1=0.85,0.95,1. At μ1=1, the solution of the proposed problem converges to the exact solution. The numerical simulations and their associated contour plot are presented to show the efficiency and stability of the numerical scheme, which can be observed from Figures 1–6. The numerical simulation of the obtained numerical solution of the Buckmaster problem is shown in Figures 7–12 for fractional order μ1=0.85,0.95,1. The last two figures reveal that the solution directly converges to the exact solution of the proposed problem. Thus, the efficiency and applicability of the suggested hybrid interpolation method is verified in these two applications.Figure 1
Numerical simulation of problem 1 forμ1=0.85.Figure 2
Contour map of numerical solution of problem 1 forμ1=0.85.Figure 3
Numerical simulation of problem 1 forμ1=0.95.Figure 4
Contour map of numerical solution of problem 1 forμ1=0.95.Figure 5
Numerical simulation of problem 1 forμ1=1.Figure 6
Contour map of numerical solution of problem 1 forμ1=1.Figure 7
Numerical simulation of problem 2 forμ1=0.85.Figure 8
Contour map of numerical solution of problem 2 forμ1=0.85.Figure 9
Numerical simulation of problem 2 forμ1=0.95.Figure 10
Contour map of numerical solution of problem 2 forμ1=0.95.Figure 11
Numerical simulation of problem 2 forμ1=1.Figure 12
Contour map of numerical solution of problem 2 forμ1=1.


## 7. Conclusion

In this study, we have devised a hybrid Newton polynomial interpolation approach to solve fractional PDEs. We have used the Caputo fractional operator. This technique combines Newton interpolation with the Laplace transform. We have derived a general numerical scheme for solution of fractional PDE using Laplace transform and Newton interpolation. The approach was verified by solving two PDEs, including fractional diffusion and Buckmaster equations. All the results have been validated through numerical simulations. In future, this method can be used to solve more difficult nonlinear fractional PDEs.



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