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A space–time spectral collocation algorithm for the variable order fractional wave equation
SpringerPlus volume 5, Article number: 1220 (2016)
Abstract
The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variableorder fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi–Gauss–Lobatto collocation scheme for the spatial discretization and the shifted Jacobi–Gauss–Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.
Background
The subject of fractional calculus is one of the branches of applied mathematics which deals with derivatives and integrals of any arbitrary order (Hilfer 2000; Kilbas and Trujillo 2002; Kilbas et al. 2006). Fractional partial differential equations are describing the phenomena in many various areas such as fluid mechanics, physics, engineering, biology (Miller and Ross 1993; Giona and Roman 1992; Rossikhin and Shitikova 1997; Podlubny 1999; West 2007). The concept of variableorder fractional allows the power of the fractional operator to be a function of the independent variable (Coimbra 2003; Chechkin et al. 2005; Evans and Jacob 2007; Sun et al. 2009; Coimbra et al. 2005; Coimbra and Ramirez 2007). Few numerical methods have been introduced and discussed to solve the variableorder fractional problems (Sun et al. 2012; Ma et al. 2012; Zeng et al. 2015; Fu et al. 2015; Abdelkawy et al. 2015). Bhrawy and Zaky (2015a) proposed a new algorithm for solving oneand twodimensional variableorder cable equations based on Jacobi spectral collocation approximation together with the Jacobi operational matrix for variableorder fractional derivative. Chen et al. (2014) proposed an implicit alternating direct method for the twodimensional variableorder fractional percolation equation also discussed the stability and convergence of the implicit alternating direct method.
Spectral methods (Canuto et al. 2006; Saadatmandi and Dehghan 2011; Doha and Bhrawy 2012; Bhrawy and Zaky 2015b, c; Bhrawy et al. 2016a) have been widely used in many fields in the last four decades. In the early times, the spectral method based on Fourier expansion has been used in few fields such as a simple geometric field and periodic boundary conditions. Recently, they have been developed theoretically and used as powerful techniques to solve various kinds of problems. Based on the accuracy and exponential rates of convergence, spectral methods have an excellent reputation when compared with others numerical methods. The expression of the problem solution as a finite series of polynomials/functions is the major step of all types of spectral methods. Then, the coefficients of this expansion will be chosen such that the absolute error is diminished as well as possible.
The spectral collocation method (Canuto et al. 2006; Bhrawy and Alofi 2013; Gu and Chen 2014; Bhrawy and Abdelkawy 2015; Bhrawy 2016a) is a specific type of spectral methods, that is more applicable and widely used to solve almost types of differential (Bhrawy et al. 2016b; Tatari and Haghighi 2014), integral (Bhrawy et al. 2016c; Rahmoune 2013), integrodifferential (Jiang and Ma 2013; Ma and Huang 2014) and delay differential (Bhrawy et al. 2015a; Reutskiy 2015) equations. While, the numerical solution will be enforced to almost satisfy the partial differential equations (PDEs) in spectral collocation method. In other words, the residuals may be permitting to be zero at chosen points. Wei and Chen (2012) proposed Legendre spectral collocation methods for pantograph Volterra delayintegrodifferential equations. Bhrawy and Alofi (2012) introduced the spectral shifted Jacobi–Gauss collocation method for solving the Lane–Emden type equation. Bhrawy et al. (2015b) proposed the spectral collocation algorithm to solve numerically some wave equations subject to initialboundary nonlocal conservation conditions in one and two space dimensions. Bhrawy (2016b) proposed Jacobi spectral collocation method for solving multidimensional nonlinear fractional subdiffusion equations.
The aim of this paper is to find the numerical solution of the space–time variable order fractional wave equation subject to initialboundary conditions. The wave equation is an important secondorder partial differential equation for the description of waves as they occur in physics such as sound waves, light waves and water waves. Variable order wave equation appears in areas such as acoustics, electromagnetics, and fluid dynamics. This paper extends the SJ–GLC and SJ–GRC schemes in order to solve the spacetime variable order fractional wave equation. The proposed collocation scheme is investigated for both temporal and spatial discretizations. The SJ–GLC and SJ–GRC are proposed, with a suitable modification for treating the boundary and initial conditions, for spatial and temporal discretizations. This treatment, for the conditions, improves the accuracy of the scheme greatly. Therefore, the space–time variable order fractional wave equation with its conditions is reduced to system of algebraic equations which is far easier to be solved. Finally, numerical examples with comparisons lighting the high accuracy and effectiveness of the proposed algorithm are presented.
The present paper is presented as follows. The definitions of the fractional calculus and some properties of Jacobi polynomials are introduced in “Preliminaries” section. The spectral collocation methods for the space–time variable order fractional wave problem subject to initialboundary conditions are presented in “Jacobi collocation method” section and then illustrated with two examples in “Numerical examples” section. The “Conclusion” is included in the last section.
Preliminaries
We first recall some definitions and preliminaries of the variableorder fractional differential and integral operators and some knowledge of orthogonal shifted Jacobi polynomials that are most relevant to spectral approximations.
Definition 1
The Riemann–Liouville and Caputo differential operators of constant order \(\gamma ,\) when \(n1\le \gamma <n,\) of f(t) are given respectively by,
where \(\Gamma (.)\) represents the Euler gamma function.
Definition 2
The left Riemann–Liouville variableorder fractional differential operator of order \(\gamma (t)\) is given by
where \(n1< \gamma _{\min }< \gamma (t)< \gamma _{\max } < n , n \in \mathbb {N}\) for all \(t \in [0,\tau ]\).
Definition 3
The Caputo variableorder fractional differential operator is given by
where \(0< \gamma (t) \le 1\) for all \(t \in [0,\tau ]\).
It is important to note here that the constantorder fractional derivative can be seen as a special case of the variableorder fractional derivative. These two definitions are related by the following relation:
The Jacobi polynomials, denoted by \(P_{j}^{(\theta ,\vartheta )}(x) (j=0,1\ldots ); \theta>1, \vartheta >1\) and defined on the interval \([1,1]\) are generated from the threeterm recurrence relation:
where
The formula that relates Jacobi polynomials and their derivatives is
The orthogonality condition is
where \(w^{(\theta ,\vartheta )}=(1x)^\theta (1+x)^\vartheta , h_{k}^{(\theta ,\vartheta )} =\dfrac{2^{\theta +\vartheta +1}\Gamma (k+\theta +1)\Gamma (k+\vartheta +1)}{(2k+\theta +\vartheta +1) k!\Gamma (k+\theta +\vartheta +1)}.\)
Let the shifted Jacobi polynomials \(P^{(\theta ,\vartheta )}_i{(\dfrac{2x}{L}1)}\) be denoted by \(P^{(\theta ,\vartheta )}_{L,i}{(x)}\), then they can be obtained with the aid of the following recurrence formula:
The analytic form of the shifted Jacobi polynomials \(P^{(\theta ,\vartheta )}_{L,i}{(x)}\) of degree i is given by
and the orthogonality condition is
where \(w_{L}^{(\theta ,\vartheta )} (x)={x}^{\vartheta }(Lx)^{\theta }\) and \(\hbar ^{(\theta ,\vartheta )}_{L,k} =\dfrac{L^{\theta +\vartheta +1}\Gamma (k+\theta +1)\Gamma (k+\vartheta +1)}{(2k+\theta +\vartheta +1) k!\Gamma (k+\theta +\vartheta +1)}\).
The shifted Jacobi–Gauss quadrature is commonly used to evaluate the previous integrals accurately. For any \(\phi \in S_{2N+1}[0,L]\), we have
where \(S_{N}[0,L]\) is the set of polynomials of degree less than or equal to \(N, x_{G ,L,j}^{(\theta ,\vartheta )}\ (0\le j \le N )\) and \(\varpi _{G ,L,j}^{(\theta ,\vartheta )}\ (0\le j \le N )\) are used as usual the nodes and the corresponding Christoffel numbers in the interval [0, L], respectively.
For shifted Jacobi–Gauss (SJ–G), \(x_{G ,L,j}^{(\theta ,\vartheta )}\ (0\le j \le N )\) are the zeros of \(P_{L,N+1}^{(\theta ,\vartheta )}(x)\) and the weights
where
while the nodes and the corresponding Christoffel numbers in the shifted Jacobi Gauss–Radau (SJ–GR) quadrature are given by \(x_{R ,L,0}^{(\theta ,\vartheta )}=0,\; x_{R ,L,j}^{(\theta ,\vartheta )}\ (1\le j \le N )\) are the zeros of \(P_{L,N}^{(\theta ,\vartheta +1)}(x),\) and the weights
A function u(x), square integrable in [0, L], may be expressed in terms of shifted Jacobi polynomials as
where the coefficients \(c_j\) are given by
The qth derivative of \(P_{L,k}^{(\theta ,\vartheta )}(x)\) can be written as
Accordingly, we can calculate the Caputo variable order derivative of shifted Jacobi polynomials from
Jacobi collocation method
In this section, we introduce a numerical algorithm extends the SJ–GLC and SJ–GRC schemes in order to solve the spacetime variable order fractional wave equation. The collocation points are selected at the SJ–GR and SJ–GL interpolation nodes for temporal and spatial variables, respectively. The core of the proposed method consists of discretizing the space–time variable order fractional wave equation to create a system of algebraic equations of the unknown coefficients. This system can be then easily solved with a standard numerical scheme.
In particular, we consider the following space–time variable order fractional wave equation
with the initial conditions
and the boundary conditions
where \(B(x,t)>0, g_{0}(x),g_{1}(x),g_{2}(t)\) and \(g_{3}(t)\) are given functions, while f(u, x, t) is a source term.
We choose the approximate solution to be of the form
where \(\mathcal {P}_{0}^{i,j,k}(x,y,t)=P_{L,i}^{(\theta _{1},\vartheta _{1})}(x)\, P_{T,j}^{(\theta _{2},\vartheta _{2})}(t).\)
The approximation of the temporal partial derivative \(D_{t}u(x,t)\) can be easily computed as follows
where \(\mathcal {P}_{1}^{i,j}(x,t)=P_{L,i}^{(\theta _{1},\vartheta _{1})}(x)\,P_{T,j}^{(\theta _{2},\vartheta _{2},1)}(t).\)
A straightforward calculation shows that the fractional derivative of variable order of the approximate solution can be computed by
where
Now, adopting (18)–(21), enable one to write (15) in the form:
while the numerical treatments of initial and boundary conditions are
In the proposed shifted Jacobi collocation method, the residual of (15) is set to be zero at \((N1)^2\) of collocation points. Moreover, the initialboundary conditions in (23) will be collocated at collocation points. Firstly, we have \((N1)^2\) algebraic equations for \((N+1)^2\) unknowns of \(\hat{u}_{i,j}\)
where
and also we have \(2(N1)\) algebraic equations which will be obtained due to the initial conditions
Furthermore, using the boundary conditions, we have \(2(N+1)\) algebraic equations
Combining Eqs. (24), (26) and (27), we obtain
The previous system of nonlinear algebraic equations can be easily solved. After the coefficients \(a_{i,j}\) are determined, it is straightforward to compute the approximate solution \(u_{N,M}(x,t)\) at any value of (x, t) in the given domain from the following equation
Numerical examples
This section reports two numerical examples to demonstrate the high accuracy and applicability of the proposed method. We also compare the results given from our scheme and those reported in the literature. The comparisons reveal that our method is very effective and convenient.
Example 1
Consider the following variable order fractional wave equation which is given in Sweilam and Assiri (2015),
where \(\beta (x,t)=1.5+0.25\cos (x)\sin (2t),\,\,\alpha (x,t)=1.5+0.5 e^{(xt)^{2}1}\) and
with the initialboundary conditions
The exact solution of this problem when \(\alpha (x,t)=\beta (x,t)=2\) is given by
Sweilam and Assiri (2015) proposed the nonstandard finite difference (NSFD) method to solve this problem with choices of \(N=1000\) and \(M=125\). In Table 1, we contrast our numerical results based on absolute errors obtained using the proposed algorithm for three choices of the shifted Jacobi parameters at \(N=8\) with the corresponding results of NSFD method (Sweilam and Assiri 2015). In Table 2, we contrast our results based on maximum absolute errors (MAEs) obtained by the present method for three choices the shifted Jacobi parameters at \(N=8\). From the results of this example, we observe that the approximate solution obtained by our method is more better than those obtained in Sweilam and Assiri (2015).
Figure 1 displays the spacegraph of the numerical solution of problem (1) with \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\). While, Fig. 2 compares graphically the curves of numerical and exact solutions of problem (1) for the different values of t at \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\). Moreover, we represent in Figs. 3 and 4 the absolute error curves obtained by the present method at \(t=0.5\) and \(x=5\) with \(N=8,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\), respectively. This demonstrates that the proposed method leads to an accurate approximation and yields exponential convergence rates.
Example 2
Consider the following problem
where
with the initial and boundary conditions
where f(x, t) is a given function such that the exact solution of this problem is
In Table 3, we list the results based on the MAEs obtained by the proposed method (with various choices of \(N, \theta _{1},\, \theta _{2},\,\vartheta _{1},\) and \(\vartheta _{2}\)). From this table, we see that we can achieve an excellent approximation for the exact solution by using proposed method for a limited number of the collocation nodes. Also this demonstrates that the proposed method provides an accurate approximation and yields exponential convergence rates.
Figure 5 shows the space graph of the absolute errors with \(N=20,\) and \(\theta _{1}=\theta _{2}=\frac{1}{2},\,\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\). While, Fig. 6 compares graphically the curves of numerical and exact solutions of problem (2) for the different values of t at \(N=20, \theta _{1}=\theta _{2}=\frac{1}{2},\,\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\). Meanwhile, we plot in Fig. 7 the absolute error curve obtained by the present method at \(t=0.5\) with \(N=20,\) and \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\). Moreover, we present in Fig. 8 the logarithmic graphs of MAEs (i.e., \(log_{10} M_E\)) obtained by the present method with different values of \((N=2,4, 6, \cdots , 20)\) at three cases of \(\theta _{1},\,\theta _{2},\,\vartheta _{1},\) and \(\vartheta _{2}\)

1.
Case 1, \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=0\).

2.
Case 2, \(\theta _{1}=\theta _{2}=\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\).

3.
Case 3, \(\theta _{1}=\theta _{2}=\frac{1}{2}, \,\vartheta _{1}=\vartheta _{2}=\frac{1}{2}\).
All the above numerical simulations demonstrate the high accuracy and applicability of our technique.
Conclusions
We presented a collocation method to achieve an accurate numerical solution for variableorder fractional wave problem subject to initialboundary conditions. One of the most advantages of the present technique is that a fully spectral method was implemented for the time and space variables by using SJ–GRC and SJ–GC approximations respectively. The problem with its conditions was then reduced to an algebraic system. The greatest feature of the present scheme is, adding few terms of the SJ–G and SJ–GR collocation points, a full agreement between the approximate and exact solutions was achieved. Through the numerical examples and specially the comparison between the obtained approximate solution and those obtained by other approximations, we demonstrate the validity and high accuracy of the present method.
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Authors’ contributions
The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.
Acknowledgements
This article was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are very grateful to the reviewers for their comments and suggestions which have improved the paper.
Competing interests
The authors declare that they have no competing interests.
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Keywords
 Variableorder fractional derivative
 Collocation method
 Jacobi polynomials
 Gauss quadrature
 Fractional wave equation