On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Mahmut Modanli 1  and Ali Akgül 2
  • 1 Harran University, 63300, Şanlıurfa, Turkey
  • 2 Siirt University, 56100, Siirt, Turkey
Mahmut Modanli
  • Harran University, Faculty of Arts and Sciences, Department of Mathematics, 63300, Şanlıurfa, Turkey
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and Ali Akgül
  • Corresponding author
  • Siirt University, Art and Science Faculty, Department Of Mathematics, 56100, Siirt, Turkey
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Abstract

The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders α = 1.1, 1.5, 1.9. Numerical results confirm the accuracy and effectiveness of the technique.

Introduction and Preliminaries

Fractional differential equations have many implementations in finance, [1, 2, 3]. These type engineering, physics and seismology equations are solvable with restpect to variables time and space. Some difference schemes are given for the space-fractional heat equations in [4, 5, 6, 7, 18, 19, 20, 21, 22]. A new difference scheme for time fractional heat equation based on the Crank-Nicholson method has been presented in [5]. Orsingher and Beghin [14] have presented the Fourier transform of the fundamental solutions to time-fractional telegraph equations of order 2α. In [15], the time-fractional advection dispersion equations have been presented. In [16], Liu has studied fractional difference approximations for time-fractional telegraph equation. Modanli and Akgül [12] have worked the second-order partial differential equations by two accurate methods. Finally, Modanli and Akgul [13] have solved the fractional telegraph differential equations by theta-method. For more details see [23, 24, 25, 26, 27].

In this study, the Crank-Nicholson difference schemes method has been applied to fractional derivatives to get numerical results.

Now, we examine the following fractional telegraph equations

{2u(t,x)t2+α1u(t,x)tα12u(t,x)x2+u(t,x)=f(t,x),0<x<L,0<t<T,1<α<2u(0,x)=r1(x),ut(0,x)=r2(x),0tT,u(t,XL)=u)t,XR)=0,XL<x<XR.

Here, r1(x), r2(x) are smooth function defined with the space [0,T ], f (t,x) is smooth function defined with the space (0,L) × (0,t) and u(t,x) is unknown function with the domain [0,L] × [0,T ]. For the equation (1), the Crank-Nicholson finite difference scheme method is applied. With using this method, obtained numerical results are very good and efficient for given examples.

Definition 1

The Caputo fractional derivative Dtαu(t,x) (t,x) of order α with respect to time is defined as:

αu(t,x)tα=Dtαu(t,x)=1Γ(nα)0t1(tp)αn+1αu(p,x)pαdp,(n1<α<n)
and for α = n ∈ N defined as:
Dtαu(t,x)=αu(t,x)tα=nu(t,x)tn.

Definition 2

First-order approach difference method for the computation of the problem (1) has been presented as:

DtαUnkgατj=0k1bj(α)(UnkjUnkj1),
where gα,τ=τ2αΓ(3α) and bj(α)=(j+1)2αj2α.

Next section, we shall give Crank-Nicholson difference scheme for fractional order telegraph differential equation.

1 Crank-Nicolson Difference Scheme and its Stabilty

Using the formula (3) and definition of Crank-Nicholson first order difference schemes, we can construct the following difference scheme formula for (1) as:

{unk+12unk+unk1τ2+gα,τj=0k1bj(α)(unkjunkj1)+12(unk+1+unk)1h2((un+1k+12unk+1+un1k+1)+(un+1k2unk+un1k)=fnk,fnk=f(tk,xn),1<α<2,un0=r1(xn),un1un0τ=r2((xn),0nM,u0k=uMk=0,0kN.

The formula (4) can be rewriten as:

{(12h2)un+1k+(12h2)un+1k+1+(1τ2)unk1+(2τ2+1h2+12)unk+(1τ2+1h2+12)unk+1+(12h2)un1k+(12h2)un1k+1+12(unk+1+unk)+gα,τj=0kbj(α)(unkjunkj1)τgα,τbkr2(xn)=fnk,1<α<2,fnk=f(tk,xn),un0=r1(xn),un1un0τ=r2((xn),0nM,u0k=uMk=0,0kN.

We can write the above system in matrix form as

Aun+1+Bun+Cun1=ϕn,
where un=[un0,un1,,unN],ϕn=[ϕn0,ϕn1,,ϕnN]T and ϕnk=f(tk,xn)+τgα,τbkr2(xn). Here A(N+1)×(N+1) and B(N+1)×(N+1) are the matrices of the following form,
A=12h2[000000011000000001000000],

B=[1000ab0gα,τb+gα,τc0b1gα,τa+gα,τ(b1b0)b+gα,τ0bk1gα,τ(bk2bk3)gα,τ(bk3bk4)gα,τcbkgα,τ(bkbk1)gα,τ(bk1bk2)gα,τb+b0gα,τ]
where a=1τ2,b=2τ2+1h2+12 and c=1τ2+1h2+12.

Then we have

un=αn+1un+1+|βn+1,1kM.

Next we should determine the matrices αn+1 and β n+1 above. Using the Dirichlet boundary conditition

u(0,t)=u(0,L)=0,0tT,
we obtain u0 = α1u1 + β 1. From that, we can choose α1 = O(N+1)×(N+1) and β1 = ON+1). Substitute un = αn+1un+1 + β n+1 and un−1 = αnun + β n into the equation (7), then
(A+Bαn+1+Aαnαn+1)un+1+(Bβn+1+Aαnβn+1+Aβn)=ϕn.

Thus, we get

A+Bαn+1+Aαnαn+1=0,Bβn+1+Aαnβn+1+Aβn=ϕn.

Thus, we obtain the following equalities

αn+1=(B+Aαn)1A,βn+1=(B+Aαn)1(ϕnAβn),
where 1 ≤ n ≤ M.

For the stability, implementing the technique of analyzing the eigenvalues of the iteration matrices of the schemes.

Let ρ(A) be the spectral radius of a matrix A, which indicates the maximum of the absolute value of the eigenvalues of the matrix A. We can write the following results.

Theorem 1

The difference scheme (5) is stable.

Proof

From the method [15], we should prove that ρ(αn) < 1, 1 ≤ n ≤ M.

ρ(α1)=0<1isclearly.ρ(α2)=B1AB1A=1min1kN1{|akk|mk,m=1N1|akm|}A=|12h2|+|12h2||1τ2+1h2+12+τ1αΓ(3α)|=1h22τ2+1h2+12+τ1αΓ(3α)1,for12+1h2τ2αΓ(3α)2τ20..

If ρ(αn) < 1, let us calculate ρ(αn+1).

A=[00000001h22τ2+1h2+12+τ1αΓ(3α)12h2αn2,21000000000000001h22τ2+1h2+12+τ1αΓ(3α)12h2αnN+1,N+1].

We know that αni = ρ(αn) and 0 ≤ ρ(αn) < 1 for 2 ≤ i ≤ N + 1. Then, we can obtain that ρ(αn+1) < 1. As a result, we obtain the desired result with induction.

Remark 2

Using Matlab programming for N = M = 10, α = 1.5,0 ≤ t ≤ 1, 0 ≤ x ≤ π and h=πM, tau=1N, we obtain the following spectral radius of a matrix as:

ρ (α1) = 0, ρ(α2) = 0.0482, ρ(α3) = 0.0484, ρ(α4) = 0.0485, ρ(α5) = 0.0482, ρ(α6) = 0.0487, ρ(α7) = 0.0487, ρ(α8) = 0.0475, ρ(α9) = 0.0487 and ρ(α10) = 0.0486.

These final results prove the stability estimation of the Theorem 1.

Remark 3

Applying the method in [16, 17], we can get the convergence of the method from stability and consistency of the proposed method.

Now, we give numerical applications for the fractional telegraph partial differential equation by Crank-Nicholson method.

2 Numerical implementation

Example

We take into consideration the following fractional telegraph partial differential equation:

{2u(t,x)t2+α1u(t,x)tα12u(t,x)x2+u(t,x)=sinx(6t+6t4αΓ(5α)+2(t3+1))0<x<π,0<t<1,1<α<2u(0,x)=sinx,ut(0,x)=0,0t1,u(t,0)=u)t,π)=0,0xπ.

The exact solution is given as u(t,x) = (t3 + 1)sinx. We implement difference schemes method to solve the problem. We utilize a procedure of modified Gauss elimination method for difference equation (8). We obtain the maximum norm of the error of the numerical solution by:

ε=max|u(t,x)u(tk,xn)|n=0,1,,Mk=0,1,2,,N
where unk=u(tk,xn) is the approximate solution. The error analysis in Table 1 gives our error analysis for difference schemes method.

Table 1

Error Analysis

τ=1N,h=piM
The difference scheme (8)In method [13]
α
N = M = 401.50.00400.0242
N = M = 801.55.4707 × 10−40.0118
N = M = 1601.50.00220.0058
N = 100, M = 101.17.0178 × 10−4
1.50.0045
1.90.0093
N = 225,M = 151.13.4496 × 10−4
1.50.0040
1.90.0083
N = 400,M = 201.12.0762 × 10−4
1.50.0034
1.90.0079

We have compared Crank-Nicholson finite difference scheme method by the theta method [13] for the variable values N = M = 40,80,160. From these comparisons, we see that this method is more effective then the method used in [13].

3 Conclusion

In this work, stability estimates were presented for fractional telegraph differential equations. Stability inequalities were given for the difference schemes method. We applied the difference schemes-method for investigating fractional telegraph partial differential equations. Approximate solutions were obtained by this method. MATLAB software program was utilized for all results.

References

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    I. I. Gorial. Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative. Eng. and Tech. Journal, 29:709–715, 2011.

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    L. Su, W. Wang, Z. Yang. Finite difference approximations for the fractional advection-diffusion equation. Physics Letters A., 373:4405–4408.2009.

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    C. Tadjeran, M. M. Meerschaert, H. P. Scheffler. A Second-order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213:205–213, 2006.

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    Zhao, Z. and Li, C. (2012) Fractional Difference/Finite Element Approximations for the Time-Space Fractional Telegraph Equation. Applied Mathematics and Computation,219, 2975–2988.

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    Ford, N.J., Rodrigues, M.M., Xiao, J. and Yan, Y. (2013) Numerical Analysis of a Two-Parameter Fractional Telegraph Equation. Journal of Computational and Applied Mathematics, 249, 95–106.

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    Orsingher, Enzo, and Luisa Beghin. “Time-fractional telegraph equations and telegraph processes with Brownian time.” Probability Theory and Related Fields 128.1 (2004): 141–160.

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    Liu, Ru. “Fractional Difference Approximations for Time-Fractional Telegraph Equation.” Journal of Applied Mathematics and Physics 6.01: 301 2018.

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    K. Nishimoto; An essence of Nishimoto’s Fractional Calculus, Descartes Press Co. 1991.

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    A. Yokus, Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics BVol. 32, No. 29, 1850365 2018.

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    A. Yokus. D. Kaya, Numerical and exact solutions for time fractional Burgers’ equation, J. Nonlinear Sci. Appl., 10 3419–3428 2017.

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    A. Yokus. D. Kaya, Conservation laws and a new expansion method for sixth order Boussinesq equation, AIP Conference Proceedings 1676, 020062 2015.

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    A. Yokus., Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 2085–2091 2018.

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    A. Yokus., H. Bulut, On the numerical investigations to the Cahn-Allen equation by using finite difference method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(1), 18–23, 2018.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C. Celik, M. Duman. Crank-Nicholson method for the fractional equation with the Riezs fractional derivative. Journal of computational physics, 231:1743–1750, 2012.

  • [2]

    I. I. Gorial. Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative. Eng. and Tech. Journal, 29:709–715, 2011.

  • [3]

    H. Jafari, V. D. Gejii. Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition. Appl. Math. and Comput., 180:488–497, 2006.

  • [4]

    I. Karatay, S. R. Bayramoglu, A. Sahin. Implicit difference approximation for the time fractional heat equation with the nonlocal condition. Applied Numerical Mathematics, 61:1281–1288, 2011.

  • [5]

    I. Karatay, S. R. Bayramoglu, A. Sahin. A new difference scheme for time fractional heat equation based on the Crank-Nicholson method. Fractional Calculus and Applied Analysis, 16:892–910, 2013.

  • [6]

    L. Su, W. Wang, Z. Yang. Finite difference approximations for the fractional advection-diffusion equation. Physics Letters A., 373:4405–4408.2009.

  • [7]

    C. Tadjeran, M. M. Meerschaert, H. P. Scheffler. A Second-order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213:205–213, 2006.

  • [8]

    Li, C. and Cao, J. (2012) A Finite Difference Method for Time-Fractional Telegraph Equation. Mechatronics and Embedded Systems and Applications (MESA), 2012.IEEE/ASME International Conference on. IEEE, 314–318.

  • [9]

    Zhao, Z. and Li, C. (2012) Fractional Difference/Finite Element Approximations for the Time-Space Fractional Telegraph Equation. Applied Mathematics and Computation,219, 2975–2988.

  • [10]

    Ford, N.J., Rodrigues, M.M., Xiao, J. and Yan, Y. (2013) Numerical Analysis of a Two-Parameter Fractional Telegraph Equation. Journal of Computational and Applied Mathematics, 249, 95–106.

  • [11]

    Sevimlican, A. (2010) An Approximation to Solution of Space and Time Fractional Telegraph Equations by He’s Variational Iteration Method. Mathematical Problemsin Engineering, 1–11 2010.

  • [12]

    Modanli, Mahmut, and Ali Akgül. “On solutions to the second-order partial differential equations by two accurate methods.” Numerical Methods for Partial Differential Equations 34.5 1678–1692 2018.

  • [13]

    Modanli, Mahmut, and Ali Akgül. “Numerical solution of fractional telegraph differential equations by theta-method.” The European Physical Journal Special Topics 226.16–18 3693–3703. 2017

  • [14]

    Orsingher, Enzo, and Luisa Beghin. “Time-fractional telegraph equations and telegraph processes with Brownian time.” Probability Theory and Related Fields 128.1 (2004): 141–160.

  • [15]

    Smith, Gordon D. Numerical solution of partial differential equations: finite difference methods. Oxford university press, 1985.

  • [16]

    Liu, Ru. “Fractional Difference Approximations for Time-Fractional Telegraph Equation.” Journal of Applied Mathematics and Physics 6.01: 301 2018.

  • [17]

    Richtmyer, Robert D., and Keith W. Morton. “Difference methods for initial-value problems.” Malabar, Fla.: Krieger Publishing Co.,| c1994, 2nd ed. 1994.

  • [18]

    A. Ashyralyev, M. Modanli. Nonlocal boundary value problem for telegraph equations. AIP Conference Proceedings, 1676:020078–z1–020078-4, 2015.

  • [19]

    S. Samko, A. Kibas, O. Marichev. Fractional Integrals and derivatives: Theory and Applications. Gordon and Breach, London, 1993.

  • [20]

    I. Podlubny. Fractional Differential Equations, “Mathematics in Science and Engineering V198”, Academic Press, San Diego 1999.

  • [21]

    K. Nishimoto; An essence of Nishimoto’s Fractional Calculus, Descartes Press Co. 1991.

  • [22]

    K.M. Kowankar, A.D Gangal; Fractional Differentiability of nowhere differentiable functions and dimensions, CHAOS V.6, No. 4, 1996, American Institute of Phyics.

  • [23]

    A. Yokus, Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics BVol. 32, No. 29, 1850365 2018.

  • [24]

    A. Yokus. D. Kaya, Numerical and exact solutions for time fractional Burgers’ equation, J. Nonlinear Sci. Appl., 10 3419–3428 2017.

  • [25]

    A. Yokus. D. Kaya, Conservation laws and a new expansion method for sixth order Boussinesq equation, AIP Conference Proceedings 1676, 020062 2015.

  • [26]

    A. Yokus., Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 2085–2091 2018.

  • [27]

    A. Yokus., H. Bulut, On the numerical investigations to the Cahn-Allen equation by using finite difference method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(1), 18–23, 2018.

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