# Numerical Simulations for the Space-Time Variable Order Nonlinear Fractional Wave Equation

Authors: Nasser Hassan Sweilam; Taghreed Abdul Rahman Assiri
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The  explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered. The concept of variable order fractional derivative is considered in the sense of Caputo. The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presented.


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## Body

## 1. Introduction

It is well known that the fractional calculus definitions are extensions of the usual calculus definitions [1–8], where the orders need not to be positive integers. On the other hand, the variable order calculus is a natural extension of the constant order (integer or fractional) calculus. In this sense, the order may function in any variable such as time and space variables or a system of other variables [9, 10]. In general, one can say that this extension is introduced by Samko and Ross in [11], where Marchaud fractional derivative and Riemann-Liouville derivative are extended to the variable order cases the order in this case is a function in the space variable only. Many authors have introduced different definitions of variable order differential operators, each of these with a specific meaning to suit desired goals. These definitions such as Riemann-Liouville, Grünwald, Caputo, Riesz [3, 12–16], and some notes as Coimbra definition [17, 18].Coimbra in [17] used Laplace transform of Caputo’s definition of the fractional derivative as the starting point to suggest a novel definition for the variable order differential operator. Because of its meaningful physical interpretation, Coimbra’s definition is better suited for modeling physical problems. The variable order differentials are an important tool to study some systems such as the control of nonlinear viscoelasticity oscillator (for more details see [17–19] and the references cited therein), where the order changes with respect to a parameter or more parameters.In the following, we present the basic definition for the variable order fractional derivatives which we will use in this paper.Definition 1 (see [14]).
The Caputo space variable order derivative is defined as follows:(1)Dx∝(x,t)u(x,t)=1Γ(n-∝(x,t))×∫0x1(x-ξ)∝(x,t)-n+1∂nu(ξ,t)∂ξndξ,
where 0<∝(x,t)<1.The main aim of this work is to use the explicit finite difference method (EFDM) to study numerically  the following nonlinear space-time variable order wave equation:(2)∂β(x,t)u(x,τ)∂τβ(x,t)=B(x,t)∂∝(x,t)u(x,τ)∂x∝(x,t)+f(u,x,t),1<∝(x,t),β(x,t)≤2,
subject to initial conditions
(3)u(x,0)=φ1(x),ut(x,0)=φ2(x),
and the following boundary conditions
(4)u(0,t)=Ψ1(x),u(a,t)=Ψ2(x),
where 0≤x<a,0≤t<T,B(x,t)>0 is a constant,  Ψ1(x),Ψ2(x) are smooth functions, and f(u,x,t) is a nonlinear scour term that satisfies the Lipschitz condition, that is,
(5)|f(u1,x,t)-f(u2,x,t)|≤L|u1-u2|,
where the constant L>0 is called a Lipschitz constant for f.


## 2. Discretization for EFDM

In this section, EFDM is used to study the model problem (2), then the space-time solutions domain will be discretized. The discrete form for the pervious Caputo derivative can be written as follows:
(6)Dx∝(x,t)u(x,t)=1Γ(2-∝(x,t))×∫0x1(x-ξ)∝(x,t)-2+1∂2u(ξ,t)∂ξ2dξ=1Γ(2-∝(x,t))∑k=0i-1∫kh(k+1)hz1-∝(x,t)×∂2u(x-z,τ)∂z2dz,z=x-ξ≈1Γ(2-∝(x,t))×∑k=0i-1({(h2)-1}(u(x-(k-1)h,t)-2u(x-kh,t)+u(x-(k+1)h,t))(h2)-1)×∫kh(k+1)hz1-∝dz.
Then,
(7)Dx∝(x,t)u(x,t)≈h2-∝(x,t)Γ(3-∝(x,t))×∑k=0i-1({(h2)-1}(u(x-(k-1)h,t)-2u(x-kh,t)+u(x-(k+1)h,t))(h2)-1)×((k+1)2-∝(x,t)-k2-∝(x,t)).
Now, pick two positive integers N, M and define the step size of space and time by h, τ, respectively, where h=a/M and τ=T/N. Also we introduce the following notations:
(8)xi=ih,fori=1,2,…,N,tj=jτ,forj=1,…,M,uij≈u(xi,tj), Bij=B(xi,tj), and fij=f(uij,xi,tj). Then,
(9)Dx∝(x,t)u(x,t)=h-αijΓ(3-αij)∑k=0i-1(ui-k+1j-2ui-kj+ui-k-1j)×((k+1)2-αij-k2-αij).
By the same way, we have
(10)Dtβ(x,t)u(x,t)=τ-βijΓ(3-βij)∑k=0j-1(uij-k+1-2uij-k+uij-k-1)×((k+1)2-βij-k2-βij).
For simplicity, let us define
(11)Rij=Bijh-∝ijΓ(3-∝ij),Qij=Γ(3-βij)τ-βij,Gkj=((k+1)2-∝ij-k2-∝ij),Hik=((k+1)2-βij-k2-βij).
Then, we can rewrite (2) in the following form:
(12)∑k=0j-1(uij-k+1-2uij-k+uij-k-1)Hik≈QijRij∑k=0i-1(ui-k+1j-2ui-kj+ui-k-1j)Gkj+Qijfij,
that is,
(13)uij+1=2uij-uij-1-∑k=1j-1(uij-k+1-2uij-k+uij-k-1)Hik+QijRij∑k=0i-1(ui-k+1j-2ui-kj+ui-k-1j)Gkj+Qijfij,uij+1=(2-Hi1)uij-∑k=2M-2(Hik-2-2Hik-1+Hik)uij-k+1-(Hij-2-2Hij-1)ui1-Hij-1ui0+QijRij∑k=0i-1(ui-k+1j-2ui-kj+ui-k-1j)Gkj+Qijfij.
The previous equation can be expressed in the following matrix form:
(14)Ui0=∅1,Ui1=Ui0+τ∅2,
and for j≥2(15)Uij+1=AjUij-∑k=2M-2(Hik-2-2Hik-1+Hik)Uij-k+1-(Hij-2-2Hij-1)Ui1-Hij-1Ui0+Fj,
where Fj=(Qijf(um-1j,xm-1,tj),…,Qijf(u1j,x1,tj))T,Uj=(uM-1j,uM-2j,…,u1j)T,
(16)∅1=(φ1(x1),φ1(x2),…,φ1(xN))T,∅2=(φ2(x1),φ2(x2),…,φ2(xN))T,
and Aj=(anmj) is a matrix with the following coefficients:
(17)anmj={QnjRnjGn-1j,wherem=1,QnjRnj(Gn-mj-2Gn-m+1j+θGn-m+2j),wherem≤n,2-Hnj+QnjRnj(θG1j-2G0j),wherem=n+1,QnjRnjG0j,wherem=n+2,0,wherem>n+2,θ={0,wherem=2,1,otherwise,
for  n=1,2,…,K-1, and m=1,2,…,K-1. Also, we note that
(18)∥A∥∞=max1≤n≤K∑m=1K|anm|=max1≤n≤K{2-Hin}=2-Hi0,
then ∥A∥∞=1.Lemma 2.
The coefficientsGkj and Hik satisfy the following conditions:(1)
G0j=1, and Hi0=1,(2)
Gkj>Gk+1j, and Hik>Hik+1, for k=0,1,…


## 3. The Stability Analysis and the Truncation Error

Let us considerWj+1 and Uj+1 to be two different numerical solutions of (15) with initial values given by W0 and U0, respectively.Theorem 3.
The explicit method approximation defined by (15) to the variable order space-time wave equation (2) is unconditionally stable, that is,
(19)|Wj+1-Uj+1|≤C|W0-U0|,foranyj.Proof.
Let us defineWj+1-Uj+1=εj+1.
From (15) we have
(20)εij+1=Ajεij-∑k=2M-2(Hik-2-2Hik-1+Hik)εij-k+1-(Hij-2-2Hij-1)εi1-Hij-1εi0+Fεj,
where
(21)Fεj=(Qm-1jf(um-1j,xm-1,tj)-Qm-1jf(wm-1j,xm-1,tj),…,Q1jf(u1j,x1,tj)-Q1jf(w1j,x1,tj))T≤(Qm-1jLm-1jεm-1j,…,Q1jL1jε1j)T=ΔFjεj,
and ΔFj=diag(Qm-1jLm-1j,…,Q1jL1j)T.
Noting that|Lij|≤L, for any i,j.
LetQ-=max{Qm-1j,…,Q1j}. From (20), we have ∥Aj+ΔFj∥m≤(2+Q-L), then
(22)∥εij+1∥∞≤∥Aj+ΔFj∥∞∥εij∥∞+∑k=2M-2(Hik-2-2Hik-1+Hik)∥εij-k+1∥∞+(Hij-2-2Hij-1)∥εi1∥∞+Hij-1∥εi0∥∞.
Now, we analyze the stability via mathematical induction [10]. From (14) we have ∥εi1∥∞≤C∥εi0∥∞, where C is a constant.
Now, assume that∥εij∥∞≤C∥εi0∥∞, then from (22), we have
(23)∥εij+1∥∞≤C(2+Q-L)∥εi0∥∞+∑k=2M-2(Hik-2-2Hik-1+Hik)C∥εi0∥∞+C(Hij-2-2Hij-1)∥εi0∥∞+Hij-1∥εi0∥∞≤C1∥εi0∥∞.
Then, the theorem holds.Lemma 4.
Let(24)0D¯x∝(xi,tj)u(xi,tj)=h-∝(xi,tj)Γ(3-∝(xi,tj))×∑k=0j-1Gij(ui-k+1j-2ui-kj+ui-k+1j)
be a smooth function; then
(25)|D-x∝(xi,tj)u(xi,tj)-Dx∝(xi,tj)u(xi,tj)|=O(h).Proof.
In terms of standard centered difference formula, we have(26)0D¯x∝(xi,tj)u(xi,tj)=h2-∝(xi,tj)Γ(3-∝(xi,tj))×∑j=0k-1Gij[∂2u(x-jh,t)∂z2+O(h2)]=h2-∝(xi,tj)Γ(3-∝(xi,tj))×∑j=0k-1Gij∂2u(x-jh,t)∂z2+h2-∝(xi,tj)k2-∝(xi,tj)Γ(3-∝(xi,tj))O(h2)=h2-∝(xi,tj)Γ(3-∝(xi,tj))×∑j=0k-1Gij∂2u(x-jh,t)∂z2+x2-∝(xi,tj)Γ(3-∝(xi,tj))O(h2)=h2-∝(xi,tj)Γ(3-∝(xi,tj))×∑j=0k-1Gij∂2u(x-jh,t)∂z2+O(h2).
By the integral mean value theorem, we have
(27)0Dx∝(xi,tj)u(xi,tj)=1Γ(2-∝(xi,tj))×∑j=0k-1∫jh(j+1)hz1-∝(xi,tj)∂2u(x-z,t)∂z2dz=h2-∝(xi,tj)Γ(3-∝(xi,tj)),
where  ξj∈[jh,(j+1)h]. Combining the pervious two formulae, we have
(28)|D-x∝(xi,tj)u(xi,tj)-Dx∝(xi,tj)u(xi,tj)|=|h2-α(xi,tj)Γ(3-α(xi,tj))×∑j=0k-1Gij[∂2u(x-jh,t)∂z2-∂2u(x-ξj,t)∂z2]+O(h2){∑j=0k-1Gij}|=|h2-α(xi,tj)Γ(3-α(xi,tj))∑j=0k-1Gij·O(h)+O(h2)|=|h2-α(xi,tj)k2-α(xi,tj)Γ(3-α(xi,tj))·O(h)+O(h2)|=O(h)+O(h2)=O(h).
Now, by using Lemma 4, we can derive the truncation error of explicit finite difference scheme (14). It has a local truncation error of O(τ) (from the left side) and O(h) (from the right side).Remark 5.
The  pervious explicit method was shown to be stable. This method is consistent with a local truncation error which isO(τ)+O(h). Therefore, according to the Lax Equivalence Theorem [2], it converges at this rate.


## 4. Numerical Examples

Example 1.
Consider the following variable-order linear fractional wave equation:(29)∂β(x,t)u(x,t)∂τβ(x,t)=-0.5cos(∝(x,t)π2)×∂∝(x,t)u(x,t)∂x∝(x,t)+f(u,x,t),
with ∝(x,t)=1.5+0.5e-(xt)2-1,β(x,t)=1.5+0.25cos(x)sin(2t), and
(30)f(u,x,t)=2ut2+1-(t2+1)×(16x2-∝(x,t)Γ(3-∝(x,t))+6x3-∝(x,t)Γ(4-∝(x,t))),
subjected to the following initial conditions:
(31)u(x,0)=φ(x)=x2(8-x),ut(x,0)=Ψ(x)=0,
where        Xa=0,  Xb=8, and T=1.The exact solution of this problem whenβ=2 is u(x,t)=x2(8-x)(t2+1).In Figure1, a comparison between the numerical and the exact solutions when β=2 at t=0.416 is presented.Figure 1In Figures2(a) and 2(b), we report the approximate solutions at t=0.052 and t=0.78, respectively.(a)
(b)In Figures3, 4, and 5, respectively, we report the approximate solutions in three dimensions, where the axis’s are (t,x,u),(alfa,x,u), and (beta,x,u), respectively.Figure 3Figure 4Figure 5Example 2.
Consider the following variable-order nonlinear fractional wave equation:(32)∂β(x,t)u(x,t)∂tβ(x,t)=2cost∂∝(x,t)u(x,t)∂x∝(x,t)+f(u,x,t),
with     ∝(x,t)=2-cos2(x)sin2(t),β(x,t)=1.8+0.5e-(xt)2-1 and       u(x,0)=φ(x)=1+sinx,ut(x,0)=Ψ(x)=0, where        0≤x≤10,T=1,  and  f(u,x,t)=u2-sin2(x)-cos2(t).
This problem has the following exact solution, when∝=2(33)u(x,t)=sinx+cost.In Figure6(a), we report the numerical solution when ∝(x,t), β(x,t) are variables at t=0.52 and the exact solutions when ∝=β=2.(a)
(b)
(c)In Figures6(b) and 6(c), we report the approximate solution at t=0.052 and t=0.78,  respectively.Figures7, 8, and 9 show the approximate solution in three dimensions, where the axes are (t,x,u), (alfa,x,u) and (beta,x,u), respectively.Figure 7Figure 8Figure 9


## 5. Conclusions

In this paper, numerical studies using a simple explicit FDM for solving the variable order space-time wave equation are presented. The stability analysis and the truncation error of the proposed method are proved. Some test examples are given, and the results obtained by the method are compared with the exact solutions in integer order cases. Several figures are presented to simulate the solutions behaviors when the variable orders change with respect to space and time. The comparison certifies that FDM gives good results. Summarizing these results, we can say that the finite difference method in its general form gives reasonable calculations, easy to use, and can be applied for the variable order differential equations in general form. All results were obtained by using MATLAB version 7.6.0 (R2008a).



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