key: cord-0782377-1dgmcmwa authors: Tuan, N.H.; Ganji, R.M.; Jafari, H. title: A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel date: 2020-09-01 journal: Chin J Phys DOI: 10.1016/j.cjph.2020.08.019 sha: 2b2044d4a3a80de8da35e1873ff1f95bca21352b doc_id: 782377 cord_uid: 1dgmcmwa In the recent years, few type of fractional derivatives which have non-local and non-singular kernel are introduced. In this work, we present fractional rheological models and Newell-Whitehead-Segel equations with non-local and non-singular kernel. For solving these equations, we present a spectral collocation method based on the shifted Legendre polynomials (SLPs). To do this, we extend the unknown functions and its derivatives using the shifted Legendre basis. These expansions and the properties of the SLPs along with the spectral collocation method will help us to reduce the main problem to a set of nonlinear algebraic equations. Finally, The accuracy and efficiency of the proposed method are reported by some illustrative examples. The study of fractional calculus started at the end of the seventeenth century. It is a branch of mathematical analysis in which integer order derivatives and integrals extend to a real or complex number [1, 2] . In the end of nineteenth century basic theory of fractional calculus was developed with the studies of Liouville, Grünwald, Letnikov, and Riemann. It has been shown that fractional derivative operators are useful in describing dynamical processes with memory or hereditary properties such as creep or relaxation processes in viscoelastoplastic materials [3, 4] , impact problem [5] , plasma physics [1] , diffusion process models [6, 7, 8, 9] , chaotic systems [10] , control problems [11, 12] , dynamics modeling of coronavirus (2019-nCov) [13] , etc. Since in definition of the most important fractional operators such as Riemann-Liouville (RL) and Caputo exists a kernel of type local and sinqular, it is difficult or impossible to describe many non-local dynamics systems. Hence novel definitions for fractional integral and derivative operators have been introduced such as Caputo-Fabrizio (CF) [14] and Atangana-Baleanu (AB) operators [15] . The most important advantage of these operators is the existence of the non-local and nonsinqular kernel which introduced to describe complex physical problems [16, 17, 18, 19, 20, 21, 22, 23] . The AB and CF derivatives show crossover properties for the meansquare displacement, while the RL derivative is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the CF kernel has a steady state between the transition. Only the AB kernel is a crossover for the waiting time distribution from stretched exponential to power law. The CF derivative is less noisy while the fractional AB derivative provides an excellent description, due to its Mittag Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state [24, 25] . Orthogonal basis functions have been generally used to achieve approximate solution for many problems in various fields of science. Approximation of the solution using these functions is known as a useful tool in solving many classes of equations, numerically, e.g., differential equations [26, 27] , integro-differential equations [28, 29] and partial differential equations [30] of various orders (fixed, fractional or variable order). In this section, many definitions of new fractional operators together with their important properties are recall which will be used further. The RL-integral of order ω satisfies the following property Definition 2.2 (See [15] ). Let 0 < ω ≤ 1, ε ∈ H 1 (0, 1) and AB(ω) be a normalization function suchthat AB(0) = AB(1) = 1 and AB(ω) = 1 − ω + ω Γ(ω) . Then i. The Caputo AB-derivative is defined as is the Mittag-Leffler function. ii. The AB-integral is given as (1) Let α ω = 1−ω AB(ω) and β ω = 1 AB(ω)Γ(ω) , then we can rewrite (1) as It is easy to report that the AB-integral satisfies the following properties [32] AB where B(·, ·) is the Beta function. where δ is a constant number. Theorem 2.2. Suppose that f and g satisfy the assumptions of Theorem 2.1, then we have Proof. According to definition of the AB-integral, we have Taking ε = α ω + β ω , the proof is complete. [14] ). Let ω ∈ (0, 1], ε ∈ H 1 (0, 1) and CF (ω) be a normalization function suchthat CF (0) = CF (1) = 1 and CF (ω) = 2 2−ω . Then i. The CF-derivative is defined as ii. The CF-integral is given as Let α ω = 2(1−ω) (2−ω)CF (ω) and β ω = 2ω (2−ω)CF (ω) , then we can rewrite (3) as The CF-integral satisfies the following properties Proof. According to definition of the CF-integral, we have Taking ε = α ω + β ω , the proof completes. The SLPs on the interval [0, 1] are defined by L n (t) = L n (2t − 1), n = 0, 1, 2, · · · , where L n (t) is the well-known Legendre polynomial (LP) of degree n. The recursive formula of LP on [−1, 1] is given by L 0 (t) = 1, L n+1 (t) = 2n + 1 n + 1 t L n (t) − n n + 1 L n−1 (t), n = 1, 2, 3, · · · . The given SLPs (L n (t)) in the equation (5), could be written the following analytic form where For two arbitrary functions h, p ∈ L 2 [0, 1] the inner product and norm in this space are defined, respectively, by For the SLPs, the orthogonality condition is as follows Suppose that ε(t) ∈ L 2 [0, 1]. Then, the function ε(t) can be expanded in terms of the SLPs by where By taking only the first M + 1 terms in (8), ε(t) can be approximated as where C = [ε 0 , ε 1 , · · · , ε M ] T and Assume ε M is the best approximation of ε into H, then the error bound is as Proof. Suppose that P M is the interpolating polynomials to ε at points t i , where t i , i = 0, 1, · · · , M are the roots of (M + 1)-degree shifted Chebysheve polynomials on [0, 1]. Then Since ε M is the best approximation of ε in H, we get By taking the squared root from both sides (12), the proof completes. Similarly, any function ε( where C = [c i,j ] is an (M + 1) × (M + 1) matrix whose elements are given by where θ 1 , θ 2 and θ 3 are positive constants. If ε M (x, t) = L T (x)CL(t) be the best approximation of ε into H * , then Proof. Let that P M is the interpolating polynomials to ε at points (x i , t j ), where x i , i = 0, 1, · · · , M and t j , j = 0, 1, · · · , M are the roots of (M + 1)-degree shifted Chebysheve polynomials on [0, 1]. Then Since ε(x, t) is a smooth function on I, then there exist constants θ 1 , θ 2 and θ 3 , such that By substituting (15) into (14) and employing the estimates for Chebysheve interpolation nodes, we have Since ε M is the best approximation of ε in H * , that is where ε * is any arbitrary polynomial in H * . Then, using (16) we obtain Finally, taking the square root of both sides of (17) completes the proof. This subsection is devoted to introducing some operational matrices (OMs) of the SLPs basis vector which will be used further. (1) The OM of the integration of the vector L(t) given by (11) can be approximated as where P is given as [29] (2) The OM of AB-integral of order ω of the vector L(t) is obtained as Now, we must obtain the OM of RL-integral of order ω. To do this, we apply the LR-integral operator, RL I ω t , on L i (t), i = 0, 1, · · · , M as By approximating the function t r+ω in terms of the SLPs, we have In view of (20) and for i = 0, 1, · · · , M , we get Therefore, for i = 0, 1, · · · , M , we can write where with ρ i,k,r = Γ(r + 1)ς i,r e r,k Γ(r + ω + 1) . By substituting (21) into (19) , the proof completes. (3) The OM of CF-integral of order ω of the vector L(t) is obtained as In view of (18), we have where I is an (M + 1) × (M + 1) identity matrix and I ω = α ω I + β ω P . The matrix I ω is called the OM of CF-integral based on the SLPs. The behavior of linear viscoelastic materials can be described by linear differential equations. In general, a constitutive equation for a linear viscoelastic material is given by , · · · , n, j = 0, 1, · · · , m, and n ≥ m. Thus, models including connected common mechanical elements such as springs and dampers can be used to visualize the constitutive equation in a convenient way. These descriptions are known as rheological models constructed by combining linear springs and dampers in series and parallel. Three well-known rheological models called Kelvin-Voigt, Maxwell, and Zener models. More complex rheological models with more realistic responses can be constructed by including additional elements [2] . A Kelvin-Voigt element is composed of a linear spring and damper connected in parallel, and its constitutive equation is given as where E is the elasticity modulus and η is the viscosity. Under a creep test with σ(t) = σ 0 and ε(0) = 0, the response is obtained as A Maxwell element is composed of a linear spring and damper connected in series, and its constitutive equation is given as Under a creep test with σ(t) = σ 0 and ε(0) = 0, the response is A Zener element is composed of a linear spring and a Maxwell element connected in parallel, and its constitutive equation is given as Under a creep test with σ(t) = σ 0 and ε(0) = 0, the response of the model is ii. Fractional approach In general, consider the following FDE: where 0 < ω ≤ 1 and λ i ∈ R + , i = 1, 2, 3. D ω t is denoted either the AB ( ABC D b. The constitutive equation of the fractional Maxwell model is obtained when c. The constitutive equation of the fractional Zener model is obtained when , ω = v. All the previous settings for λ i yield the classic rheological models when ω = v = 1. iii. The method In here, we introduce a numerical method for the solution of the form equation (23) . Let in the equation (23), the derivative is described in the AB (or CF) sense. For solving the equation (23), first we approximate ABC D ω t ε(t) and ABC D v t σ(t) as By taking the AB-integral of (24) and using initial conditions (ε(0) = 0, σ(0) = σ 0 ), we have By approximating σ(0) C T 3 L(t), σ(t) can be rewritten as where By putting (24)- (26) in (23), we have Figures 1 and 2 , when η = 0.5 and σ 0 = 1 for the AB and CF derivatives, respectively. We used Mathematica for computation. i. Classic approach Consider the Newell-Whitehead-Segel equation and boundary conditions where λ 1 , λ 2 and λ 3 are real numbers with λ 1 > 0, and λ 4 is a positive integer number. When λ 1 = 1, λ 2 = 1, λ 3 = 1 and λ 4 = 2, the equation (28) is called the Fishers equation. ii. Fractional approach Consider the Newell-Whitehead-Segel equation by where 0 < ω ≤ 1. In the fractional Newell-Whitehead-Segel equation (31) , D ω t is called the AB or CF derivative operator. iii. The method Here, we introduce a numerical method for the solution of the form equation (31) . Let in the equation (31), the derivative is described in the AB (or CF) sense. For solving the equation (31), first we approximate D xx ε(x, t) as where C 1 is an (M + 1) × (M + 1) unknown vector. By integrating from (32) respect x twice, we get ε(x, t) L T (x) P 2 T C 1 L(t) + xε x (0, t) + ε(0, t). By putting x = 1 into (33), we have where the elements of C 2 , C 3 , C 4 and C 5 vectors can be calculused by (9) . With helping (34) and (35), ε(x, t) can be rewritten as where . We approximate F and ε(x, 0) using the shifted Legender basis by where C 7 is an (M +1)×(M +1) unknown vector and the elements of C 8 vector can be obtained by (9) . By taking the AB-integral of both sides of the equation (31) and using (32) , (36), (37) and (38) , we get Now, by putting (36) into (37) and using the collocation points x i = i M +2 , i = 1, 2, . . . , M + 1 and t j = j M +2 , j = 1, 2, . . . , M + 1, gives Equations (39) and (40) form a system of 2(M + 1)(M + 1) nonlinear equations of the vectors of C 1 and C 7 . By solving this system, the unknown parameters of the vectors of C 1 and C 7 are obtained. Finally the approximate solution can be computed by (36). Consider the Newell-Whitehead-Segel equation in the following cases Case 1. By considrting the values λ 1 = 1, λ 2 = 1, λ 3 = 1, λ 4 = 2, the Newell-Whitehead-Segel equation with the exact solution ε(x, t) = 1 (1+e is as follows is as follows t is as follows In this work, we have presented a numerical method for solving fractional rheological models and Newell-Whitehead-Segel equations. The derivative is considered in the Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) sense. Our numerical method is based on the operational matrices (OMs) of the shifted Legendre polynomials (SLPs). By this way, the main problem is reduced to a system of nonlinear algebraic equations which greatly simplifies the problem. An error estimate is proved for the approximate solution. Finally, some examples have been presented to demonstrate the accuracy and efficiency of the proposed method. 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Jafari On Behalf of authors All authors discussed the results and contributed to the final manuscript This research received no external funding. The authors declare no potential conflict of interests.