key: cord-0057937-xmn2su9f authors: Edwan, Reem; Saadeh, Rania; Hadid, Samir; Al-Smadi, Mohammed; Momani, Shaher title: Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method date: 2020-11-24 journal: Computational Mathematics and Applications DOI: 10.1007/978-981-15-8498-5_2 sha: 1cd6288d4df0b7e34dc817e6712cc80f1a05e78c doc_id: 57937 cord_uid: xmn2su9f In this chapter, we present the time-space-fractional Cauchy equation with constant coefficients, the space and time-fractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. The implicit scheme is introduced to solve time-space-fractional Cauchy problem in a matrix form by utilising fractionally Grünwald formulas for discretization of Riemann-Liouville fractional integral, and L1-algorithm for the discretization of time-Caputo fractional derivative, additionally, we provided a proof of the von Neuman type stability analysis for the fractional Cauchy equation of fractional order. Several numerical examples are introduced to illustrate the behaviour of approximate solution for various values of fractional order. used to model issues in finance, viscoelasticity, mathematical biology and chemistry [1] [2] [3] [4] [5] [6] [7] [8] . Different partial differential equations of fractional order are studied and resolved by several powerful methods [9] [10] [11] [12] [13] [14] [15] [16] [17] . Consequently, considerable attention has been given to the answer of partial differential equations of fractional order. Several powerful strategies are established and developed to induce numerical and analytical solutions of fractional differential equations, like finite-difference technique [7] , finite volume technique [9] , finite element technique [11] , homotopy perturbation technique [13] and the fractional sub-equation technique [2] . Recently, many scholars introduced methods for solving fractional differential equations. Momani developed a domain decomposition technique to approximate solution for the fractional convection-diffusion equation with a nonlinear source term [12] . Dehghan et al introduced a numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multi-wavelets [8] . Saadatmandi et al studied the sinc-Legendre collocation technique for a category of fractional convection-diffusion equations with variable coefficients [14] . Liu et al introduced the finite volume technique for solving the fractional diffusion equations [18] , and Yang et al proposed the finite volume technique to the fractional diffusion equations [19] , all of that are without theoretical analysis. Meerschaert and Tadjeran proposed the finite-difference technique for the resolution of the fractional advection-dispersion flow equations [16] . Baeumer and Meerschaert obtained the solution for fractional Cauchy equations by subordinating the solution of the original Cauchy equation [20] . Pskhu introduced a fundamental solution of a higher order Cauchy equation with time-fractional derivative [21] . Recently, Hejazi et al utilised the finite volume technique and finite-difference technique for solving the space-fractional advection-dispersion equation [17] . They used fractionally shifted Grünwald formula for the fractional derivative and verified the stability and convergence of the scheme, whose order is O(τ + h). During this Chapter, we propose a finite-difference technique to get a new approximate solution for the time-space-fractional Cauchy equation with constant coefficients, space-fractional derivative and time-fractional derivative are described within the Riemann-Liouville sense and Caputo sense, respectively. Consider the time-space-fractional Cauchy equation of the shape subject to the initial condition is a given function provided that u(x, t), g(x, t), and f (x) are smooth enough, is a positive parameter, β is a parameter describing the order of the space fractional, and γ is a parameter describing the order of the time-fractional, the space-fractional derivative and timefractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. The starting point for a finite-difference discretization is a partition of the computational domain [a, b] into a finite number of sub-domains V i , i = 0, 1, 2, . . . , N , known as control volumes CVs, the union of all CVs should cover the whole domain. We introduce the implicit scheme by discretization of the Riemann-Liouville fractional integral, and time-Caputo fractional derivative. For another numerical scheme, see [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] . This chapter introduces a finite-difference technique for solving the time-spacefractional Cauchy equation with constant coefficients and contains the following sections: Sect. 2 is devoted to mathematical preliminaries. The description of a modified finite-difference technique is presented in Sect. 3. The von Neuman type stability analysis and consistency are proved in Sect. 4. Whilst the numerical experiments are given in Sect. 5. Finally, a brief conclusion is outlined in the last section. Throughout the past decade, fractional calculus has been applied to virtually every field of engineering, economics, science and another field. People like Liouville, Riemann and Weyl created major contributions to the idea of fractional calculus [32] [33] [34] [35] [36] [37] [38] [39] . The story of the fractional calculus continued with contributions from Fourier, Abel, Leibniz, Grünwald and Letnikov. Over the years, several definitions found that are acceptable for the concept of fractional derivatives and integrals [40] [41] [42] [43] [44] [45] [46] [47] [48] . is the identity operator. Let n ∈ N be the smallest integer that exceeds α, then the Riemann-Liouville fractional derivative of order α > 0 is defined by provided that D α a u(x) = D n J (n−α) where a < x ≤ b, with h = x−a N is called the Grünwald-Letnikov fractional derivative of order α of the function u. Definition 4 Let n be the smallest integer that exceeds α, then the Caputo fractional derivative of order α > 0 is defined by ]. The following theorem shows the relation between this definition and the Riemann-Liouville fractional derivatives: . . , then we may rewrite (5) as This formula is used to approximations the fractional integrals J α a u(x). Lemma 1 Let 0 < α < 1. Then, we have Proof For the first part, let w α 0 = 1 and w α 1 = α > 0, thus from the recursive definition and since 0 < α < 1, we have 0 (7) is strictly between zero and one. Now, the second part can be done for j ≥ 2 such that Finally, from 1 and 2 we have for j ≥ 2 Whenever we use a numerical technique to solve a differential equation, we would like to make sure that the numerical solution obtained is a sufficiently good approximation to the actuality solution, some necessary definition and remarks are introduced to discuss the stability analysis [27, [49] [50] [51] [52] [53] [54] . To analyse the stability of difference scheme for IVP, suppose that we are given a vector in 2 . .) T , and define the discrete fourier transform of v as follows: of the discrete scheme. For simplification, we can get the discrete Fourier transform of the difference scheme by replacing v n j in the difference scheme by v n j = v n ex p(i jξ ), i = √ −1. The difference scheme v n+1 = Qv n is stable with respect to 2,h norm if and only if there exist positive constants τ 0 , h 0 and so that |ρ Remark 3 If ρ satisfies the inequality in Remark 2, then ρ is said to be satisfied the von Neumann condition. The difference scheme that is stable under a set of conditions is called conditionally stable, otherwise is called unconditionally stable scheme. In this section, we propose a new finite-difference method for solving the time-spacefractional Cauchy equation of the shape: subject to the initial condition u( is a given function provided that u(x, t), g(x, t), and f (x) are smooth enough, is a positive parameter, β is a parameter describing the order of the space fractional and γ is a parameter describing the order of the time-fractional, the space-fractional derivative and time-fractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. Using the definition of Riemann-Liouville fractional derivative where 0 < β ≤ 1, we have where we approximate the α order fractional Riemann-Liouville integral with standard Grünwald formula and approximate the first derivative with central difference formula: A finite-difference discretization is applied by evaluating Eq. (9) at x = x i , and using the above equations. Let t n = nτ , n = 0, 1, 2, . . ., where τ is the time step, and discretise the Caputo time-fractional derivative using L1-algorithm, Collecting like terms, we can rewrite Eq. (13) as: where i = 0, 1, . . . , N , Denoting the numerical solution vector U n = u n 0 , u n 1 , . . . , u n N and source vector g n+1 = g(x 0 , t n+1 ), g(x 1 , t n+1 ), . . . , g(x N , t n+1 ) , we have the following vector equation: where the matrix A has elements a i j = b i j . In particular, for γ = 1 we can use the standard backward difference to approximate the time derivative in Eq. (12) du(x i , t) dt yields the numerical solution Anyhow, we can rewrite Eq. (16) as a vector equation in the form where the matrix A has elements a i j = b i j . In the next section, we prove that this scheme is conditionally stable, and it is first-order accurate in time and second-order accurate in space. In this section, the stability analysis for the proposed numerical scheme is presented as in the following theorems: The symbol of numerical scheme is , and satisfies the von Neumann condition if By using Reverse Triangle Inequality, we have got (16) is consistent with second-order accuracy in direction of space and first order in direction of time. Proof By using Eqs. (10), (11) and (15), we can write Eq. (8) at (x i , t n+1 ) as follows: Thus, we get that which is Eq. (8) at (x i , t n+1 ), γ = 1. The numerical scheme (14) is consistent with second-order accuracy in space and 2-γ order in time. Proof we can write Eq. (8) at (x i , t n+1 ) as follows: which is Eq. (8) at (x i , t n+1 ), 0 < γ < 1. In this section, in order to solve the fractional Cauchy equation using the finitedifference discretization scheme (FDDS), the equation is presented in a discrete specific form. Anyhow, we consider four illustrated examples to demonstrate the performance and efficiency of the proposed algorithm. The computations are performed by Wolfram-Mathematica software 11. subject to the initial condition where = 1 × 10 −3 , t ≥ 0, x ∈ [1, 4], γ = 1, and 0 < β ≤ 1. In particular, the exact solution of IVPs (18) and (19) at β = 1, γ = 1 is given by u(x, t) = sin(π (x − t)). Following the FDD algorithm, using h = 0.05 and τ = 0.01, the numerical results of FDDS with varying fractional order β such that β ∈ {0.75, 0.85, 0.95, 1}, γ = 1, compared with exact solution are given in Table 1 at the time t = 0.5 and x ∈ [1, 1.5]. In the light of showing the agreement between the FDDS and exact solutions, the absolute error of IVPs (18) and (19) are listed in Table 2 for β = 1, γ = 1 when t = 0.5 and x ∈ [1, 1.5] with h = 0.1. Table 3 is devoted to the FDDS approximate solutions at β = 0.95, γ = 1 with varying times t such that t = 0.5 and t = 1.0 over the interval [1, 1.5] with h = 0.05. From these tables, it can be noted that the FDDS approximate solutions are in good agreement with the exact solutions over the domain of interest. Anyhow, more iteration leads to more accurate solutions. For further analysis, the 2D-plot of the FDDS and exact solution for Example 1 are drawn in Fig. 1 at t = 0.5 and x ∈ [1, 3 .5]. Whilst, the surface plot of the approximate solution at β = 0.95, γ = 1 is shown in Fig. 2. Example 2 Consider the following non-homogeneous fractional Cauchy equation: subject to the initial condition where g(x, t) = sin(2x), = 2, t ≥ 0, x ∈ [0, 0.9], β = 0.85, and 0 < γ ≤ 1. Following the FDD algorithm, using h = 0.05 and τ = 0.025, the numerical results of FDDS with varying fractional order γ such that γ ∈ {0.75, 0.85, 0.95, 1}, β = 0.85 are given in Table 4 at the time t = 0.5 and x ∈ [0, 0.5]. Table 5 is devoted to the FDDS approximate solutions at β = 0.85 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] with h = 0.05, the 2D-plot of the FDDS for Example 5.2 is drawn in Fig. 3 at t = 0.5 and x ∈ [0, 0.9]. Figure 4 shown the FDDS approximate solutions at β = 0.85 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] . Whilst, the surface plot of the approximate solution at β = 0.85, γ = 1 is shown in Fig. 5 at t = 1.0. where = π , t ≥ 0, x ∈ [0, 0.95], β = 0.75, and 0 < γ ≤ 1. Following the FDD algorithm, using h = 0.05 and τ = 0.025, numerical results of FDDS with varying fractional order γ such that γ ∈ {0.75, 0.85, 0.95, 1}, β = 0.75 are given in Table 6 at the time t = 0.5 and x ∈ [0, 0.5]. Table 7 is devoted to the FDDS approximate solutions at β = 0.75 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] with h = 0.05, the 2D-plot of the FDDS for Example 5.3 is drawn in Fig. 6 at t = 0.5 and x ∈ [0, 0.95]. Figure 7 shown the FDDS approximate solutions at β = 0.75 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval subject to the initial condition In particular, the exact solution of IVPs (24) and (25) at β = 1 is given by u(x, t) = e ξ (x− t) . Following the FDDS algorithm, using h = 0.0625 and τ = 0.01, the numerical results of the exact and FDDS for different values of fractional order β such that β ∈ {0.75, 0.85, 0.95, 1}, γ = 1 are given in Table 8 at the time t = 0.5 and x ∈ [−2, −1.25]. Table 9 is devoted to the FDDS approximate solutions at β = 0.95 with varying times t such that t = 0.5 and t = 1 over the interval [−2, −1.5] with h = 0.0625. Figure 11 shown the FDDS approximate solutions at β = 0.95 and γ = 1.0 with varying times t such that t = 0.5 and t = 1 over the interval [−2, 1] . Whilst, the surface plot of the FDDS approximate solution at β = 0.95, γ = 1 is shown in Fig. 12 . From these graphs, it can be concluded that In this chapter, a new finite-difference technique has been developed for solving linear Cauchy equation of fractional order. We introduce the implicit scheme by discretization of the space-Riemann-Liouville fractional integral, and time-Caputo fractional derivative, the solution obtained using this technique shows that this approach can solve the problem effectively. The basic idea of this approach can be further utilised to resolve the linear Cauchy equation of fractional order with a variable coefficient or apply the finite volume method by using the same discretization. Funding This research was funded by Ajman University, UAE (Grant ID 2020-COVID 19-08: GL: 5211529). 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