A class of developed schemes for parabolic integro differential equations

International Journal of Computer Mathematics ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/gcom20 A class of developed schemes for parabolic integro-differential equations D Rostamy & F Mirzaei To cite this article: D Rostamy & F Mirzaei (2021): A class of developed schemes for parabolic integro-differential equations, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2021.1901278 To link to this article: https://doi.org/10.1080/00207160.2021.1901278 Published online: 19 Mar 2021 Submit your article to this journal Article views: View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gcom20 INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS https://doi.org/10.1080/00207160.2021.1901278 ARTICLE A class of developed schemes for parabolic integro-differential equations D Rostamy a and F Mirzaeib a Department of Mathematics, Imam Khomeini International University, Qazvin, Iran; b Department of Mathematics, Islamic Azad University South Tehran Branch, Tehran, Iran ABSTRACT ARTICLE HISTORY In this paper, we propose a class of methods to solve the parabolic Volterra integro-differential equations with bounded and unbounded domains More precisely, we change the parabolic Volterra integro-differential equations to well-posed linear and nonlinear dynamical systems Then, the obtained systems are solved by using a new class of algorithms consisting linear multi-step formulas in which these schemes are constructed through the hybrid of Gergory’s formula, finite difference and multi-step methods Error bounds are derived in both bounded and unbounded domains Some numerical examples are then presented to illustrate the efficiency and accuracy of the proposed methods Furthermore, stability and convergence of proposed methods are established and we denote the numerical simulations Moreover, some tests are conducted on data with measurement noise to consider the performance of the proposed methods Received April 2020 Revised October 2020 Accepted 25 February 2021 KEYWORDS Parabolic Volterra integral equations; integro-differential equations; partial differential equations 2010 MATHEMATICS SUBJECT CLASSIFICATIONS 45K05; 65M06; 65N12 Introduction In this paper, we consider the following initial-boundary-value problem for the one-dimensional diffusion equation in bounded and unbounded domains with memory term ⎧ t m ⎪ ⎨ut (x, t) + k(x, t − τ )u(x, τ ) dτ − (u (x, t))xx = f (x, t), (x, t) ∈ × [0, T], (1) x∈ , u(x, 0) = u0 (x), ⎪ ⎩ u(x, t) = 0, (x, t) ∈ ∂ × [0, T], where ∂ is the boundary of and m = 1, 2, (we denote that (1) for m = is a nonlinear diffusion equation with memory (see [37] and references therein)) The functions f, k and u0 are known and u is unknown function We assumed that f is continuous and satisfies the Lipschitz condition on × [0, T] Moreover, the function f (x, t), u0 (x) have compact support, k is a differentiable and L2 function with respect to both its variables such that ⊆ R These parabolic Volterra integro-differential equations (PVIDEs) are a class of very important evolution equations which they describe in many of physical phenomena including heat conduction for material with memory, compression of poro-viscoelastic media and nuclear reactor dynamics (e.g [7] and references therein) Recently, many efforts have been devoted to the investigation of numerical solution for different kinds of parabolic Volterra integro-differential equations such as finite element methods [7], time discretization via Laplace transformation (q.v [23]), two splitting positive definite mixed finite element methods [15] and a finite difference scheme is proposed [33] CONTACT D Rostamy rostamy@khayam.ut.ac.ir © 2021 Informa UK Limited, trading as Taylor & Francis Group D ROSTAMY AND F MIRZAEI In the papers [14] and [35], we observe that H1-Galerkin nonconforming mixed finite element and Least-squares Galerkin finite element methods are purposed for another form of PVIDEs, respectively Also, the hybrid of finite central difference and an hp-version discontinuous Galerkin (dG) with finite element approximations are used for a special case of linear PVIDEs (see [20,24]) A posteriori error analysis of the Crank-Nicolson finite element method for the linear PVIDEs is introduced in [28] On the other hand, the spectral method has been also proposed for a special case of linear PVIDEs [10] A discussion related to existence, uniqueness and asymptotic behaviour of the solution has been provided for a special case of linear PVIDEs in [9] and A-Stable linear multi-step methods and convergence analysis for some algorithms have been studied in [4,22] In the unbounded domain, a numerical solution is provided by Han et al [17,18] and finite element methods are studied by [21] Han et al (see in [6,36] and the references therein) proposed the exact absorbing boundary conditions ABCs for another nonlinear PVIDEs Brunner et al in [5] investigated the artificial boundary methods ABMs for other nonlinear PVIDEs on the unbounded spatial domain Different numerical methods for solving PVIDEs are presented by many authors, e.g [1,8,26,29–31] The present methods are hybrid of finite difference method and extended approaches for PVIDEs in [4] The following motivations arise when studying (1) are as follows (1) We need to investigate the existence and uniqueness of solution of linear and nonlinear form of (1) where they have not been proven so far (2) The schemes outlined here provide the ability to analyse conditional stability and convergence analysis for bounded and unbounded domain In addition to being extremely accurate rater than spectral methods that this class of methods is a multi-purpose shot based on standard techniques (3) Implementations of these proposed schemes are very easy rater than finite element methods (4) The main advantages to hybrid Gergory’s formula, finite difference and multi-step methods (HGFDM) are (i) It requires less computational time to obtain approximation solution with good accuracy (ii) It reduces computational cost due to the use of this method (iii) We use these methods for linear and nonlinear PVIDEs in both bounded and unbounded domains with high order accuracy The paper is structured as follows In Section 2, we introduce some definitions for existence and uniqueness of solution of (1) and the approximation solution idea In Section 3, we develop a multistep method for PVIDEs on bounded and unbounded domains based on dynamical systems The convergence analysis is provided and we show that these systems are well-posed and then we use stable algorithms for discretization In Section 4, we investigate several numerical examples to illustrate the performance and efficiency of the proposed methods Also, more detailed comparison of the results with the spectral method mentioned in[10] is given in this section Finally, we end in Section with some concluding remarks Existence and uniqueness solution To develop the method presented in [4] for PVIDEs, it integro-differential equation of the form ⎧ ⎪ ⎨ut (x, t) = Fm (x, t, u(x, t), z(x, t)), u(x, 0) = u0 (x), ⎪ ⎩ u(x, t) = 0, is convenient to rewrite (1) as a Volterra (x, t) ∈ × [0, T], x∈ , (x, t) ∈ ∂ × [0, T], (2) INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS where ⎧ m ⎪ ⎨Fm (x, t, u(x, t), z(x, t)) := (u (x, t))xx + f (x, t) − z(x, t), t z(x, t) := K(x, t − τ , u(x, τ ))dτ , ⎪ ⎩ K(x, t − τ , u(x, τ )) := k(x, t − τ )u(x, τ ) Let the time step length be denoted by k and k = T/N with N ∈ N and a subscript n to the time level tn = nk, n = 0, 1, , N For the approximations of u(x, tn ) and z(x, tn ), we use the un (x) and zn (x), respectively Like as [4], we use by M = ( , σ ; ) the application to (2) of a linear multi-step method L = ( , σ ) and a class of appropriate quadrature formulae in the following manner ( , σ ): p p αν un+ν (x) = k ν=0 βν Fm (x, tn+ν , un+ν (x), zn+ν (x)), (3) ν=0 where p p ν αν r , (r) := ν=0 βν r ν , σ (r) := αk = 1, |α0 | + |β0 | = 0, ν=0 that are the first and second characteristic polynomials of the linear multi-step method (2), respectively Also, n zn (x) = k γni K(x, tn − τi , ui (x)), τi = ik, z(x, t0 ) = z0 (x) = 0, (4) i=0 where {γni } are quadrature weights To approximate z(x, tn ), we use of Gregory’s quadrature formula that back to James Gregory (1638-1675)( cf [12,16,25,27]) A major advantage in Gregory’s quadrature compared to a traditional quadrature like rectangular, trapezoidal or Crank-Nicolson rule [28] is high order of convergence Gregory’s formula [3] A trapezoidal error occurs at the end of the interval The Gregory interpolant depends only on point values This makes it readily usable in practical computation and a competitor with other schemes for interpolation of data at equally-spaced points The quadrature formulas improve the accuracy of the trapezoidal rule by adjusting the weights near the ends of the integration interval [13] Gregory’s formula of the form [2,4,32] tn 1 ϕ0 + ϕ1 + + ϕn−1 + ϕn − k (∇ϕn − 2 12 19 (∇ ϕn − ϕ03 ) + + (∇ ϕn + ϕ02 ) + 24 720 ϕ(ξ ) dξ ≈ k + cq (∇ q ϕn + (−1)q q ϕ0 ) , (n = 1, , N), ϕ0 ) (5) for n q, we consider ϕn := K(x, tn − τi , ui (x)), i = 1, , n, and ui (x) = u(x, ti ) and for q = 0, this is the well-known extended trapezoidal rule ( cf [32,34]) Gregory’s formula admit step by step improvement of accuracy by the addition of correction terms the left-hand sides in the formulation (5) The trapezoidal rule is well known to be a second-order method (k = 2) and the accuracy of Simpson is only O(k4 ), but the order of error in (5) is of the form O(kq+2 ) when h → for tn = nk (see [32]) We remind that the order of the p-step method depends on the degree of convergence of the integration method (see [4]) 4 D ROSTAMY AND F MIRZAEI In order to apply method (3), in addition to the starting values u0 (x), , up−1 (x), we also need the sufficient numbers of starting values {uμ (x)} (μ not necessarily an integer) to approximate the quadrature as follows J(ν, n) := tν K(x, tn − τ , u(x, τ )) dτ , n = 1, , N, ν = 1, , q − For n q − 1, set zn (x) = J(n, n), and for n q, set zn (x) equal to the right-hand side of (5) In the following, we modify the definitions of [4] for two dimensions Definition 2.1: With the linear multi-step method (3), we associate the linear difference operator L and M defined by L[u(x, tn ); k] := M[u(x, tn ); k] := p ν=0 p ν=0 (αν u(x, tn+ν ) − kβν ut (x, tn+ν )), (αν u(x, tn+ν ) − kβν Fm (x, tn+ν , u(x, tn+ν ), Z(x, tn+ν ))), where n = 0, , N − p The order of L is defined as the order of ( , σ ) for (1) It is easily verified that, for all sufficiently smooth functions Fm , the operators L and M are related by p M[u(x, tn ); k] = L[u(x, tn ); k] + k βν ν=0 where m = l, , N − p, also z∗ (x, t ∂Fm (x, tn+ν , u(x, tn+ν ), z∗ (x, tn+ν )) En+ν , ∂z n+ν ) lies between Z(x, tn+ν ) and z(x, tn+ν ) Definition 2.2: Let L be of order p∗ , and p-step method ( , σ ; ) by have order q∗ Then, we define the order r∗ of the linear r∗ := min(p∗ , q∗ ) Theorem 2.1: We assume the following sets S := {(x, t, τ , u) : ≤ τ ≤ t ≤ T, H := {(x, t, u, z) : ≤ t ≤ T, |u| < ∞} , |u| < ∞, |z| < ∞} Also, if we consider Fm (x, t, u, z) := um (x − h, t) − 2um (x, t) + um (x + h, t) + f (x, t) − z(x, t) h2 such that h > and we define the following hypothesis: (H1 ) Fm (x, t, u, z), Fm (x, t, u, z) ∈ C(H), K(x, t − τ , y) ∈ C(S ), (H2 ) ˜ z) Fm (x, t, u, z) − Fm (x, t, u, ∞ ≤ L1 u − u˜ ∞, ˜ z) ∈ H, ∀(x, t, u, z), (x, t, u, (H3 ) ˜ z) Fm (x, t, u, z) − Fm (x, t, u, ∞ ≤ L2 u − u˜ ∞, ˜ z) ∈ H, ∀(x, t, u, z), (x, t, u, for some Li ≥ 0, i = 1, , Therefore, if ⎧ ⎪ ⎨ut (x, t) = Fm (x, t, u(x, t), z(x, t)), u(x, 0) = u0 (x), ⎪ ⎩ u(x, t) = 0, and we assume that (x, t) ∈ × [0, T], x∈ , (x, t) ∈ ∂ × [0, T], (6) INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS (i) (ii) (iii) (iv) Fm in (1) satisfy the hypothesis H1 , and H2 , Fm in (6) satisfy the hypothesis H1 and H3 , the p-step method ( , σ ) is zero-stable, the p-step method ( , σ ; ) is of at least order one Then, (1) under the hypothesis H1 and H2 , (1) has a unique solution (2) under the hypothesis H1 and H3 , method of (6) has a unique solution and also ( , σ ; ) is convergent Proof: To prove part 1, we are essentially concerned with an ordinary differential equation, when x is fixed So, the result is proved in [4], and we omit the detailed proof we refer the interested reader to [4] for a detailed proof On the other hand, for proving part 2, our problem is related to an ordinary differential equation in (6) We know that a linear p-step method is said to be convergent if, for all equations (1) lim k→0, nk=tn un (x) = u(x, t), holds for all t ∈ [0, T], and for all solutions {un (x)} of (3) satisfying starting conditions un (k) = un (x) for which limk→0 un (k) = u0 , n = 0, , p − 1, and J(ν, ν) for which limk→0 J(ν, ν) = 0, ν = 1, , l − Here, the last condition essentially requires that the weights in the quadrature formula, used to compute the starting values J(ν, ν) remain bounded as k → An argument similar to the one used in (6) shows that this method has a unique solution and ( , σ ; ) is convergent (cf [19,34]) The following results concerning reducible analysis of A-stability for (2) were derived in ([4]) (see for example, [19]) We reminded that if the p-step method ( , σ ; ), defined by (3) and (4), is applied to the equation (6), then, we assumed that (i) ∂ Fm = ξ = constant, ∂u ∂ Fm ∂K = η = constant ∂z ∂u (7) (ii) the interval local truncation errors of integration method and of multi-step method are functions independent to x, The following definitions are derived from standard definitions for ordinary equations differential in ([4]) (see, for example, [19],p 64) Definition 2.3: A interval region of the (kξ , k2 η)-plane is said to be a interval region of absolute stability of the multi-step method, if for all (kξ , k2 η) ∈ the method multi-step is absolutely stable Definition 2.4: The method ( , σ ; ) is said to be interval A-stable if its region contains the quarter plane kξ < 0, k2 η < of absolute stability In practical applications, once the region for a particular method has been established, estimates for ξ and η are computed from (7) (such estimates can be re-evaluated from time to time as the numerical solution proceeds), and k is chosen such that (kξ , k2 η) ⊆ Implementation of HGFDM In this section, we illustrate the proposed method and give a starting procedure for its implementation for both bounded and unbounded spatial domains in two sections 6 D ROSTAMY AND F MIRZAEI 3.1 Parabolic Volterra integro-differential equations on bounded domain In this section, we formulate a hybrid method of Gergory’s formula, finite difference and the multistep methods to solve the parabolic Volterra integro-differential equations of the second kind (6) with ∂ = {−1, 1} and = (−1, 1), so that, both the derivatives and the integrals are disconnected Now, we define grid points on the rectangular regions [−1, 1] × [0, T] with the step size h and the time-step k, respectively x −x Thus, the spatial nodes are xj = x0 + jh, xj ∈ , j = 1, , J, with (h > 0, h = j J , J ∈ N) Then, we estimate the spatial derivative uxx in (6) through applying the following standard formula of numerical differentiation (um (x, t))xx ≈ m (t) − 2U m (t) + U m (t) Uj−1 j j+1 h2 , j = 2, , J − The notations Uj (t) and u0 (xj ) are used for the approximations of u(xj , t) and u(xj , 0), respectively m (t) − 2U m (t) + U m (t) Uj+1 ∂Uj (t) j j−1 − = ∂t h2 t k(xj , t − τ )Uj (τ )dτ + f (xj , t), (8) where ≤ j ≤ J − We use Gregory’s formula (5) for numerical integration of (8), to obtain a system of J−2 ordinary differential equation subject to the following initial-boundary conditions U1 (t) = UJ (t) = 0, Uj (0) = u0 (xj ), ≤ j ≤ J − (9) Then, we use multi-step methods to solve it At first, we rewrite (8) in conventional matrix form and define U(t) := [U2 (t), , UJ−1 (t)]T , m (t)]T , U m (t) := [U2m (t), , UJ−1 U0 := [u0 (x2 ), , u0 (xJ−1 )]T , and G(t) := [f (x2 , t), , f (xJ−1 , t)]T , where U, U0 , U m and G are column vectors of length J−2 Therefore, we have ⎡ ⎤ −2 · · · ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ E := ⎢ ⎥ h ⎣ ⎦ · · · −2 (J−2)×(J−2) Now consider the function Z(t) as follows t Z(t) := where ˜ − τ )U(τ ) dτ , −k(t ⎤ k(x2 , t − τ ) · · · ⎥ ˜ − τ) = ⎢ k(t ⎦ ⎣ · · · k(xJ−1 , t − τ ) (J−2)×(J−2) ⎡ INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS Using these matrices, Equations (8) and (9) can be rewritten as Ut (t) = A1 U m (t) + U(0) = U0 , (t), 0, m = 1, 2, t (10) where A1 [ ] := E[ ] and (t) = Z(t) + G(t) (It is better to use the different notations in bounded and unbounded domains, so we introduce A1 for bounded domain.) 3.2 Solutions formula and stability for (10) In this section, we investigate the well posedness of (10) Theorem 3.1: (i) If m = then the solution of the problem (10) is given by t U(t) = eA11 U0 + A e(A11 [ ](t)−A11 [ ](s)) (s) ds, j 11 where eA11 := ∞ j=0 j! is the integrating factor and A11 [ ](t) = (ii) If m = then the solution of the problem (10) is given by t A1 [ ](s) ds Z (t) = etB Z0 , where B := v(0)e t 0 −A11 −A11 U(ξ ) dξ (t) (11) (12) , Z (t) = [v(t)U(t), v(t)]T , Z0 = [v(0)U0 , v(0)]T and v(t) = is the integration factor Proof: (i) Multiplying (10) by the integrating factor e−A11 and integrate from to t to get the desired result Since dA11 Ut (t)e−A11 − U(t)e−A11 = (t)e−A11 dt Hence, we write d(U(t)e−A11 ) = (t)e−A11 dt We denote the variable by s and integrate from to t to get t d(U(s)e−A11 ) ds = ds t −A11 (s)e ds, i.e U(t)e−A11 − U(0) = t −A11 (s)e ds Now since A11 [ ](0) = therefore we get the desired results (ii) We represent the Riccati differential equation for (10) in the projective space Z = (vU, v) as a system of two first-order linear ordinary differential equations [19] Theorem 3.2: (i) Using (11), we derive the stability estimates if A11 4|t| U(t) If A11 ∞ ∞ ≤ e h2 U0 8|t| ∞ + e h2 ( Z(s) ∞ ∞ = then + G(s) ∞) (13) = then U(t) ∞ ≤ U0 ∞ + Z(s) ∞ + G(s) ∞ (14) D ROSTAMY AND F MIRZAEI (ii) Using (12), we derive the stability estimates if A11 U(t) ≤ v(0) ∞ ∞ ∞ (1 + < α and (t) ∞ U0 |t|max{α,β} ∞ )e =e A11 < β then (15) Proof: (i) First, it easy to show that ∞ eA11 ∞ j ∞ A11 j! ≤ j=0 ∞ < eα Therefore, we write U(t) A11 [ ](t) ∞ ≤ U0 ∞e ∞ ≤ U0 ∞e t + ∞ eA11 [ ](t)−A11 [ ](s) ∞ (s) ∞ ds, hence we have U(t) A11 [ ](t) ∞ On the other hand, we⎡know that −2 · · · ⎢ t ⎢ ⎢ (I)-A11 [ ](t) = ⎢ h ⎣ · · · −2 (II)- A11 [ ](t) ∞ = 4|t| , h2 (III)- A11 [ ](t) − A11 [ ](s) the inequalities (13) and (14) (ii) Second, by using (12) we have, = ∞ 4|t−s| h2 Z (t) Therefore, we know that B proof is complete ∞ ∞ t + ≤ e A11 [ ](t)−A11 [ ](s) ∞ (s) ∞ ds ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (J−2)×(J−2) 8|t| , h2 ≤ e|t| then according to (I), (II) and (III), we have B ∞ ≤ max{α, β} and Z0 Z0 ∞ ∞ ≤ v(0) ∞ (1 + U0 ∞ ), hence, the Corollary 3.1 (Stability estimates): From inequality (11), we conclude the following inequality U(t) 8|t| where κ(t, h) := e h2 and α = 4|t| h2 ∞ ≤ α ∞e U0 + κ(t, h) G(t) − κ(t, h) k(t) ∞ ∞ , (16) Also, for inequality (14) we have the following inequality: U(t) ∞ U0 ≤ ∞ + G(t) − k(t) ∞ ∞ (17) Proof: Based on Z(t) ∞ = (13) and (14), we conclude (16) and (17) t ˜ − τ )U(τ ) dτ −k(t , ∞ INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS Remark 3.1: In the same way as for (10), it follows, if U and U are solutions of (10) when m = with {U0 , Z(t), G(t), k(t)} and {U0 , Z(t), G(t), k(t)} then for (i) 4|t| =0 h2 U(t) − U(t) or for (ii) 4|t| h2 ∞ U0 − U0 ≤ 4|t| ∞e h + κ(t, h) G(t) − G(t) − κ(t, h) k(t) − k(t) ∞ , ∞ = 0, and m = 1, we have U(t) − U(t) ∞ ≤ U − U0 ∞ + G(t) − G(t) − k(t) − k(t) ∞ ∞ Also, with the above assumptions for m = 2, we have the following inequality: U(t) − U(t) ∞ ≤ (1 + U0 − U0 |t| max{α,β} ∞ )e 4|t| In Theorem 3.2, the stability estimates have the growth in the order of e h2 It seems that the estimate is not sharp by noting that the spectral radius of A1 is negative Thus, it is hard to prove the decay, but 4|t| it is not necessary to have the growth in the order of e h2 The model problems studied here seems that it holds the maximum principle Therefore, based on the above remark, we conclude the following corollary Corollary 3.2: For m = 1, (10) is conditionally well posedness for |t| < When 4|t| h2 h2 ln k(t) − k(t) 4|t| h2 = when (18) ∞ = and m = 1, we have the following condition for well posedness ≤ k(t) − k(t) ∞ (19) Proof: By using Remark 3.4 and the produced bounds in this remark, we conclude (18) and (19) respectively Finally, we observe that when m = and max{α, β} is very small near zero then (10) is conditionally well posedness In the next section, we choose a class of stable schemes 3.3 Numerical solution of (10) The numerical solution for the dynamical system (10) was computed by using two well-known multistep methods that one of them is the Adams-Moulton method of fourth-order(AM4), and the another is the Runge-Kutta method of fourth-order(RK4) Therefore, we review them for solving (10) as follows • Explicit Runge-Kutta method of fourth-order K1 = Fm (tn , U(tn ), Z(tn )), K2 = Fm tn + 1 t, U(tn ) + tK1 , Z(tn + t) , 2 10 D ROSTAMY AND F MIRZAEI K3 = Fm tn + 1 t, U(tn ) + tK2 , Z(tn + t) , 2 K4 = Fm (tn + t, U(tn ) + tK3 , Z(tn + t)), t (K1 + 2K2 + 2K3 + K4 ), U(tn+1 ) = U(tn ) + where Wn := U(tn ), n = 0, , N • Implicit Adams-Moulton method of fourth-order Wn+1 = Wn + k 9Fm,n+1 + 19Fm,n − 5Fm,n−1 + Fm,n−2 , 24 where Wn := U(tn ), Fm,n := Fm,n (tn , U(tn ), Z(tn )), for n = 0, , N Hence, we consider WN (t) as the interpolation solutions of (10) by RK4 and AM4 methods RK4 method is explicit and it can be solved easily the initial start vector U0 AM4 method is implicit, so RK4 method can be used for the first three steps 3.4 Stability and convergence analysis for RK4 and AM4 In this section, we give an error bound for problem (10) We assume that WN (t) is the approximated solution of problem (10) applying RK4 and AM4 methods Corollary 3.3: Let U(t) and WN (t) be the exact and the approximate solutions of RK4 and AM4 for problem (10), respectively Also, we assume that {U0 , G(t), k(t)} are perturbed by {δU0 , δG(t), δ k(t)} = and (18), we have then for (i) m = 1, α = 4|t| h2 U(t) − WN (t) ≤ ∞ δU0 α ∞e + κ(t, h) δG(t) − κ(t, h) δ k(t) ∞ ∞ , or for (ii) m = 1, α = and (19), we have U(t) − WN (t) ∞ ≤ δU0 ∞ + δG(t) − δ k(t) ∞ ∞ For (iii) m = and T is very small near zero, we have U(t) − WN (t) ∞ ≤ (1 + δU0 max{α,β} ∞ )e Finally for (i), (ii) and (iii) we have lim N→∞ U(t) − WN (t) ∞ = Proof: From Theorem 2.1, Corollary 3.1 and Remark 3.1, limN→∞ δU0 ∞ = 0, limN→∞ δ k(t) ∞ = and limN→∞ δG(t) ∞ = 0, we immediately conclude the assertion INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 11 3.5 Parabolic Volterra integro-differential equations on unbounded domain We consider the following problem on the unbounded domain = R as follows, ⎧ t ⎪ ⎪ ⎪ κ(y, t − τ )u(y, τ )dτ = (um (y, t))yy + f (y, t), (y, t) ∈ R × [0, T], ⎨ut + ⎪u(y, 0) = u0 (y), ⎪ ⎪ ⎩u → as |y| → ∞ y ∈ R, (20) There are several typical mappings that convert unbounded domains to the bounded domains, that algebraic, logarithmic and exponential are more practical types of mappings Now, we use the algebraic mapping r(x) : (−1, 1) −→ (−∞, +∞), with positive scaling parameter L, Lx r(x) = √ , − x2 on (20) results are shown in the following ⎧ t ∂v ⎪ ⎪ ⎪ κ r(x), t − τ v(x, τ )dτ + ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ∂ vm r (x) ∂vm ⎪ ⎪ ⎨= − + g(x, t), ∂x2 r (x) r (x) ∂x ⎪ ⎪ v(x, 0) = u0 r(x) , ⎪ ⎪ ⎪ ⎪ ⎪ v(1, t) → 0, x → 1, ⎪ ⎪ ⎪ ⎩v(−1, t) → 0, x → −1, (x, t) ∈ (−1, 1) × (0, T], (21) where v(x, t) := u(r(x), t), g(x, t) := f (r(x), t) and m = 1, Now, we transform the problem (21) into the ordinary differential equation vt (x, t) = Fm (x, t, v(x, t), z(x, t)), ≤ t ≤ T, x ∈ [−1, 1], with the initial condition (21) and ⎧ r (x) m ⎪ ⎪ (vm )(x, t))xx − (v (x, t))x + g(x, t) − z(x, t), Fm (x, t, v(x, t), z(x, t)) = ⎪ ⎪ ⎪ ⎨ r (x) r (x) t ⎪ K(x, t − τ , v(x, τ ))dτ , z(x, t) := ⎪ ⎪ ⎪ ⎪ ⎩K(x, t − τ , v(x, τ )) := k(x, t − τ )v(x, τ ) Implementing the procedure of HGFDM, we have ⎧ m (t) − 2V m (t) + V m (t) m m Vj+1 ∂Vj (t) r (x) Vj+1 (t) − Vj (t) ⎪ j j−1 ⎪ ⎨ − = ∂t h2 h (r (x))2 (r (x))3 t ⎪ ⎪ ⎩ − k(r(x ), t − τ )V (τ )dτ + g(x , t), m = 1, 2, ≤ j ≤ J − j j (22) j Where, the approximations of v(xj , t) and v(xj , 0) are denoted by Vj (t) and v0 (xj ), respectively The system (22) is J−2 an ordinary differential equation with the following initial-boundary conditions V1 (t) = VJ (t) → 0, Vj (0) = v0 (xj ), ≤ j ≤ J − 1, (23) where v0 (xj ) := u0 (r(xj )) To write (22) in conventional matrix form, we define the following vectors and matrices V(t) := [V2 (t), , VJ−1 (t)]T , m V m (t) := [V2m (t), , VJ−1 (t)]T , (24) 12 D ROSTAMY AND F MIRZAEI V0 := [v0 (x2 ), , v0 (xJ−1 )]T , ⎡ ⎢ ⎢ B=⎢ ⎢ ⎣ r (x2 )2 ··· ··· ⎡ ⎢ ⎢ ⎢ C = −⎢ ⎢ ⎣ So that V, V0 , Vm ⎤ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ 1⎢ F= ⎢ h⎢ ⎣ r (xJ−1 )2 and H(t) := [g(x2 , t), , g(xJ−1 , t)]T , , (J−2)×(J−2) −1 · · · ··· ··· (25) ⎤ ⎥ ⎥ ⎥ , ⎥ −1 ⎦ (J−2)×(J−2) ⎤ r r (x2 )2 ··· r r (xJ−1 )2 ··· ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (J−2)×(J−2) and H are column vectors of length J−2 We define t Z(t) := where ˜ − τ )V(τ ) dτ , −k(t ⎡ k(r(x2 ), t − τ ) · · · ˜ − τ) = ⎢ k(t ⎣ ··· Therefore, the matrix forms of (22) and (23) are ⎧ ⎨ Vt (t) = A2 V m (t) + ⎩ ⎤ ⎥ ⎦ k(r(xJ−1 ), t − τ ) (t), m = 1, 2, t (J−2)×(J−2) 0, (26) V(0) = V0 , where A2 [ ] = (BE − CF)[ ] and (t) = Z(t) + H(t) (26) is a dynamical system Here, we denote ξ := ρ(BE − CF) as a tridiagonal matrix and we investigate the absolute stability region in next sections For more information on these types of matrices and their specific values, see [11] 3.6 Stability and convergence analysis of proposed schemes for (26) In this section, we investigate the well posedness for (26) Theorem 3.3: The solution of the problem (26) is given by (i) for m = V(t) = eA22 V0 + t eA22 [ ](t)−A22 [ ](s) where eA22 is the integrating factor and A22 [ ](t) = (ii) for m = t e t −A22 V(ξ ) dξ −A22 (t) (27) A2 [ ](s) ds Z (t) = etB Z , where B := (s) ds, (28) , Z (t) = [v(t)V(t), v(t)]T , Z = [v(0)V0 , v(0)]T and v(t) = v(0) is the integration factor INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 13 Proof: (i) Similar to Theorem 3.1, again with multiplying (26) by the integrating factor e−A22 and integrate from to t to get the desired result Since Ut (t)e−A22 − dA22 U(t)e−A22 = dt −A22 (t)e Hence, we write d(U(t)e−A22 ) = dt −A22 (t)e We denote the variable by s and integrate from to t to get d(U(s)e−A22 ) ds = ds t t −A22 (s)e ds, i.e t U(t)e−A22 − U(0) = −A22 (s)e ds Now since A22 [ ](0) = therefore we get the desired results (ii) According to [19], we represent the Riccati differential equation for (26) in the projective space Z = (vV, v) as a system of two first-order linear ordinary differential equations Theorem 3.4: (i) Using the solution formula, i.e (27), we derive the stability estimates if ≤ A22 ∞ < β then V(t) If β = A22 ∞ ∞ ≤ V0 eβ + e2β ( Z(s) ∞ + H(s) ∞) (29) = then V(t) ≤ V0 + Z(s) ∞ ∞ + H(s) ∞ (ii) Using the solution formula, i.e (28), we derive the stability estimates if A22 (t) ∞ < β then V(t) ∞ ≤ (1 + V0 (30) ∞ < α and |t| max{α,β} ∞ )e (31) Proof: An argument similar to the one used in the proof of Theorem 3.2 shows that the claim is true Corollary 3.4 (Stability estimates): From inequality (29), we conclude the following inequality V(t) ∞ ≤ V0 β ∞e + κ(t, h) H(t) − κ(t, h) k(t) ∞ where κ(t, h) is dependent on BE−CF Also, for inequality (30), we have the following inequality V(t) ∞ ≤ V0 ∞ + H(t) − k(t) ∞ ∞ ∞ , 14 D ROSTAMY AND F MIRZAEI Corollary 3.5: Let V(t) and WN (t) be the exact and the approximate solutions of RK4 and AM4 for problem (26), respectively Also, we assume that {V0 , H(t), k(t)} are perturbed by {δV0 , δH(t), δ k(t)} then (i) if β = and (18), we have V(t) − WN (t) ∞ ≤ δV0 β ∞e + κ(t, h) δH(t) − κ(t, h) δ k(t) ∞ ∞ , or for (ii) β = and (19), we have V(t) − WN (t) ∞ ≤ δV0 ∞ + δH(t) − δ k(t) ∞ ∞ , and also for (iii) m = and T is very small near zero, we have V(t) − WN (t) ∞ ≤ (1 + δV0 max{α,β} ∞ )e Finally for (i), (ii) and (iii), we have lim N→∞ V(t) − WN (t) ∞ = Finally problem (26) is well-posed and we proved that the proposed class of schemes are stable for solving it Numerical experimental results In this section, some numerical examples based on Theorems 3.2 and 3.4 are considered to demonstrate the efficiency and accuracy of the proposed methods, both on bounded and unbounded domains For methods of reasonably high order, the derivation of stability regions in the (kξ , k2 η)plane becomes prohibitively complicated It is, of course, feasible to calculate estimates for ξ and η from the numerical solution, and to check computationally whether, with the corresponding values of kξ and k2 η, the zeros of the stability polynomial lie within the unit disk In all examples, we use numerical methods of HGFDM by using two kinds of multi-step methods AM4 and RK4 We consider Gregory’s formula (5) with q = The numerical solution proceeds, values of ξ and η are obtained from (7), or we can consider ξ the spectral radius of matrix E and (BE − CF) Also, the spectral radius of the diagonal matrix k˜ is η As we know, a small perturbation in initial data may generate a large amount of perturbations in the solution Therefore, we apply noisy data to show the stability of the proposed methods: b˜ i = bi (1 + rand(i)), i = 1, , N, where bi is the exact data and rand(i) is a random number with uniform distribution of the interval [−1, 1] The magnitude displays the noise level of the measurement data In this section, we implement some numerical results of the proposed methods on some examples, such that the theoretical results are tested by some numerical examples The exact solutions are available for these examples We test the stability and accuracy of the illustrated method to evaluate further the computational productivity of the presented method In the following examples, we compare the rate of convergence of the spectral method in [10] and HGFDM mentioned in current article In [10], the spectral method for the similar problems (1) and (20) were investigated for linear problem The required CPU times to run the program will be presented We perform our computations using Matlab 2017 software on a Core i5, 2.67 GHz CPU machine with Gbyte of memory Moreover, we give the rough proofs of the convergence order both for the linear and nonlinear cases in this section INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 4.1 Bounded domain 15 = [−1, 1] Example 4.1: Consider the problem (1) with k(x, t) = ext , and the exact solution is u(x, t) = (1 − x2 ) sin(t) If m = then xt +cos(t)+xsin(t)) f (x, t) = (1 − x2 )cos(t) + (x −1)(−e (1+x + 2sin(t), and if m = 2, we have 2) f (x, t) = (x2 − 1)(cos(t) − etx + xsin(t)) − 8x2 sin(t)2 − 4sin(t)2 (x2 − 1) − cos(t)(x2 − 1) (x2 + 1) Fm In which E = ∂∂U and its spectral radius is ξ = −4 Moreover, the spectral radius of the diagonal h2 t matrix is η = −e if we assume k = ext with x ∈ [−1, 1] The stability interval for the Adams-Moulton method of fourth order on the real axis of (−3, 0) Hence, we have kξ = −4k ≥ −3, h2 and as a result, k ≤ h2 √ Similarly, k2 η should be in the stability interval Thus,√ if we consider k ≤ 3e−t and fix step length, an upper bound for k occurs at t = T = then, k < 3e−1 After 1000 times of testing and using statistical simulation, we were able to reach this conclusion But we insist that this must be proved by mathematical reasoning As shown in the form, the Adams-Moulton method is stable for k = 0.6h2 and h = 0.2 and it is unstable for k = 0.8h2 and h = 0.2 However, it is stable with the Runge-Kutta method of fourth order Tables and show the errors of Example 4.1 by using HGFDM method with RK4 and AM4 described above for m = 1, In these tables, we are used of the maximum norm for different values of h and k The columns labelled order gives the estimated order of convergence, using Order = EN log log2 EN , (32) where EN = max|U(t) − WN (t)| By investigation in Tables and 2, we observe the following results: Figure The numerical solutions of Example 4.1 by HGFDM for h = 0.1, k = 0.4h2 and exact solution u(x, t) = (1 − x ) sin(t) 16 D ROSTAMY AND F MIRZAEI Table Results of Example 4.1 by HGFDM with AM4 and RK4, when = 0.001 and m = AM4 J 20 20 20 20 20 N 112 224 450 900 1800 E ∞ ∗ 2.58e−05 6.42e−06 1.41e−06 1.99e−07 RK4 Order CPU time(s) − 2.06 2.18 2.82 6.528 20.452 40.452 95.852 225.546 E ∞ 3.62e−03 9.06e−04 2.28e−04 5.68e−05 1.42e−05 Order CPU time(s) 1.9904 1.9905 2.0051 2.0 1.854 15.658 35.658 82.486 105.251 Table Results of Example 4.1 by HGFDM with AM4 and RK4, when = 0.001 and m = AM4 J 20 20 20 20 20 N 112 224 450 900 1800 E ∞ ∗ 8.63e−03 2.204e−03 5.51e−04 2.39e−05 RK4 Order CPU time(s) − 1.98 2.00 2.06 9.25 25.842 42.258 75.254 120.589 E ∞ 2.68e−02 8.135e−03 2.352e−03 6.57e−04 1.75e−04 Order 1.71 1.79 1.84 1.91 CPU time(s) 6.19 0.985 4.527 33.541 90.548 Table Compearing CPU time(s) for the spectral method in [10] and HGFDM for Example 4.1, when m = (J, N) (16, 16) (20, 20) (30, 30) (20, 500) (20, 1000) RK4 AM4 Spectral method 0.295 0.925 1.452 46.257 90.257 0.997 1.162 5.824 62.85 129.85 45.984 102.854 3505.658 ∗∗ ∗∗ (1) The rate of convergence improves as N increases (2) The Adams-Moulton method is more accurate than the Runge-Kutta method (3) The rate of convergence in the linear system is better than the rate of convergence in the nonlinear system (4) By increasing step size, no significant change is observed in the approximate solution But, when step size k exceeds 0.0075, the he approximate solution is not stability Not charging the solution arises from this fact that the order of method depend on the order of integration method and because the order of method is lowest order of integration and multistep method so, the order of convergence of the method is approximately four In Table 3, we compare the rate of convergence of our method with applied method in [10] for m = Which shows a great improvement in the rate of convergence for HGFDM with AM4 and RK4 We observe that CPU time for HGFDM is better than spectral methods Moreover, in all tables, ∗ denotes the error is more than the certain tolerance and ∗∗ denotes the CPU time is more than 3600 (s) 4.2 Unbounded domain =R Consider the problem (20) with the unbounded domain = R Calculating the spectral radius of the (BE − CF) matrix with using MATLAB For h = 0.2, we will have ξ = ρ(BE − CF) ≈ 84.476 INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 17 Figure (a) Comparison of the maximum absolute error between HGFDM with AM4 and RK4 when h = 0.1, k = 0.001, 0.002, 0.003, 0.004 for Example 4.1, when m = (b) The absolute error HGFDM with AM4 (c) The absolute error HGFDM with RK4 Table Results of Example 4.2 by HGFDM with AM4 and RK4, when = 0.001 and m = AM4 J N 20 20 20 20 20 112 224 450 900 1800 E ∞ ∗ 2.61e−04 6.72e−05 1.67e−05 3.89e−06 RK4 Order CPU time(s) − 1.96 2.008 2.102 5.854 34.891 48.256 90.256 123.224 E ∞ 3.85e−02 1.113e−02 3.089e−03 8.22e−04 2.08e−04 Order CPU time(s) 1.76 1.84 1.90 1.97 1.458 10.892 20.751 49.489 85.234 ≈ 0.035 Through consecutive experiments ξ we find out that the stability regions is still under discussion and investigation The results reported in Tables and Here, we assume that the mapping parameter Q = As a result, we expect the step size bounded as −k < Example 4.2: Consider the problem (20), k(x, t) = e−x t and the exact solution is u(x, t) = f (x, t) that is defined accordingly If m = 1, f (x, t) = and if m = 2, f (x, t) = cos(t) e−t x2 − cos(t) + x2 sin(t) 8x2 sin(t) sin(t) + , + − 2 x + (x + 1) (x2 + 1)(x4 + 1) (x + 1)3 4sin(t)2 (x2 +1)3 + cos(t) (x2 +1) − 24x2 sin(t)2 (x2 +1)4 + e−tx −cos(t)+x2 sin(t) (x2 +1)(x4 +1) sin(t) 1+x2 and 18 D ROSTAMY AND F MIRZAEI Table Results of Example 4.2 by HGFDM with AM4 and RK4, when = 0.001 and m = AM4 J 20 20 20 20 20 N 112 224 450 900 1800 E ∞ ∗ 2.48e−02 8.52e−03 2.84e−03 9.304e−04 RK4 Order CPU time(s) − 1.54 1.58 1.61 5.258 15.254 42.258 100.258 190.365 E ∞ 3.04e−01 1.31e−01 5.54e−02 2.28e−02 9.19e−03 Order CPU time(s) 1.21 1.24 1.28 1.31 2.96 7.456 20.581 080.687 120.854 Figure (a) Comparison of the maximum absolute error between HGFDM with AM4 and RK4 when h = 0.1, k = 0.001, 0.002, 0.003, 0.004 for Example 4.1, when m = (b) The absolute error HGFDM with AM4 (c) The absolute error HGFDM with RK4 Figure The numerical solutions of Example 4.2, when m = by HGFDM for h = 0.1, k = 0.4h2 and exact solution u(x, t) = sin(t) (1 + x ) INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 19 Figure (a) Comparison of the maximum absolute error between HGFDM with AM4 and RK4 when h = 0.1, k = 0.001, 0.002, 0.003, 0.004 for Example 4.2, when m = (b) The absolute error HGFDM with AM4 (c) The absolute error HGFDM with RK4 Table Comparing CPU time (s) for the spectral method in [10] and HGFDM for Example 4.2, when m = (J, N) (6, 20) (10, 30) (20, 40) (20, 500) (20, 1000) RK4 AM4 Spectral method 0.225 0.852 9.549 35.257 56.128 0.621 4.824 15.483 58.972 120.24 5.922 91.641 2942.454 ∗∗ ∗∗ The numerical results of solving this example and for HGFDM with AM4 and RK4 are presented in Tables and for m = and m = 2, respectively The relative errors are given with L∞ norm for different values as follow The results show that HGFDM with AM4 is more accurate than HGFDM with RK4, and the rate of convergence is more For m = 1, the results of using [10] method are presented in Table 6, we can be seen, the rate CPU tim(s) of convergence of both methods introduced in this article are better than the method introduced in [10] Also, the numerical and exact solutions are presented in Figure Conclusion In this paper, we proposed a class of numerical schemes for the linear and nonlinear parabolic Volterra integro-differential equations By using the matrix form of these equations, the problem is reduced to 20 D ROSTAMY AND F MIRZAEI Figure (a) Comparison of the maximum absolute error between HGFDM with AM4 and RK4 when h = 0.1, k = 0.001, 0.002, 0.003, 0.004 for Example 4.2, when m = (b) The absolute error HGFDM with AM4 (c) The absolute error HGFDM with RK4 the stability of solution of a class of dynamical system of algebraic equations thus greatly simplifying the problem We extend our spatially discrete methods based on [2] because we obtain the dynamical system for stability analysis We prove the stabilities of the proposed method and presented some numerical examples to illustrate the efficiency and accuracy of the proposed methods However, in the convergence and stability analysis, we introduce a different approach from the bounded domain and unbounded domain This study concerns both theoretical and numerical aspects The results are justified by some numerical implementations We have not investigated on the proposed methods suited for problems with weakly singular kernel This important issue needs to be addressed 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Mirzaeib a Department of Mathematics, Imam Khomeini International University, Qazvin, Iran; b Department of Mathematics, Islamic Azad University South Tehran Branch, Tehran, Iran ABSTRACT ARTICLE... its variables such that ⊆ R These parabolic Volterra integro- differential equations (PVIDEs) are a class of very important evolution equations which they describe in many of physical phenomena including