A class of developed schemes for parabolic integro differential equations

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International Journal of Computer Mathematics

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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 forparabolic 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.

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A class of developed schemes for parabolic integro-differential equations

D Rostamy aand 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

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

equa-tions 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

accu-racy of the proposed methods Furthermore, stability and convergence of

proposed methods are established and we denote the numerical

simula-tions Moreover, some tests are conducted on data with measurement noise

to consider the performance of the proposed methods.

ARTICLE HISTORY

Received 2 April 2020 Revised 1 October 2020 Accepted 25 February 2021

KEYWORDS

Parabolic Volterra integral equations;

integro-differential equations; partial differential equations

2010 MATHEMATICS SUBJECT

where∂ is the boundary of  and m = 1, 2, (we denote that (1) for m = 2 is a nonlinear diffusion

equation with memory (see [37] and references therein)) The functions f, k and u0are 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 importantevolution equations which they describe in many of physical phenomena including heat conductionfor 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 ofnumerical solution for different kinds of parabolic Volterra integro-differential equations such asfinite element methods [7], time discretization via Laplace transformation (q.v [23]), two splittingpositive definite mixed finite element methods [15] and a finite difference scheme is proposed [33]

CONTACT D Rostamy rostamy@khayam.ut.ac.ir

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,

respec-tively 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 teriori error analysis of the Crank-Nicolson finite element method for the linear PVIDEs is introduced

pos-in [28] On the other hand, the spectral method has been also proposed for a special case of linearPVIDEs [10]

A discussion related to existence, uniqueness and asymptotic behaviour of the solution has beenprovided for a special case of linear PVIDEs in [9] and A-Stable linear multi-step methods and con-

vergence analysis for some algorithms have been studied in [4,22] In the unbounded domain, anumerical 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 condi-tions ABCs for another nonlinear PVIDEs Brunner et al in [5] investigated the artificial boundarymethods ABMs for other nonlinear PVIDEs on the unbounded spatial domain Different numericalmethods 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 convergenceanalysis for bounded and unbounded domain In addition to being extremely accurate raterthan spectral methods that this class of methods is a multi-purpose shot based on standardtechniques

(3) Implementations of these proposed schemes are very easy rater than finite elementmethods

(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 unboundeddomains with high order accuracy

The paper is structured as follows In Section2, we introduce some definitions for existence anduniqueness of solution of (1) and the approximation solution idea In Section3, we develop a multi-step method for PVIDEs on bounded and unbounded domains based on dynamical systems Theconvergence analysis is provided and we show that these systems are well-posed and then we use stablealgorithms for discretization In Section4, we investigate several numerical examples to illustrate theperformance and efficiency of the proposed methods Also, more detailed comparison of the resultswith the spectral method mentioned in[10] is given in this section Finally, we end in Section5withsome concluding remarks

2 Existence and uniqueness solution

To develop the method presented in [4] for PVIDEs, it is convenient to rewrite (1) as a Volterraintegro-differential equation of the form

Let the time step length be denoted by k and k = T/N with N ∈Nand a subscript n to the time level

tn = nk, n = 0, 1, , N For the approximations of u(x, t n) and z(x, tn), we use the un(x) and zn(x),

respectively

Like as [4], we use byM = (, 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):

γniK(x, tn − τ i , u i(x)), τi = ik, z(x, t0) = z0(x) = 0, (4)

where{γ ni } are quadrature weights To approximate z(x, t n), we use of Gregory’s quadrature

for-mula that back to James Gregory (1638-1675)( cf [12,16,25,27]) A major advantage in Gregory’squadrature compared to a traditional quadrature like rectangular, trapezoidal or Crank-Nicolsonrule [28] is high order of convergence Gregory’s formula [3] A trapezoidal error occurs at theend of the interval The Gregory interpolant depends only on point values This makes it read-ily 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 theform [2,4,32]

 t n

0ϕ(ξ) dξ ≈ k

1

2ϕ0+ ϕ1+ + ϕ n−1+1

2ϕn − k

1

for nq, we consider ϕn:= K(x, t n − τ i , u i(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 stepimprovement of accuracy by the addition of correction terms the left-hand sides in the formula-tion (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(k q+2) when h → 0 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])

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{u μ(x)} (μ not necessarily an integer) to approximate the

quadrature as follows

J(ν, n) :=t ν

0 K(x, tn − τ, u(x, τ)) dτ, n = 1, , N, ν = 1, , q − 1.

For nq − 1, set z n(x) = J(n, n), and for nq, 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 operatorL

andMdefined by

L [u (x, tn); k] :=p ν=0 (αν u (x, tn +ν ) − kβν ut(x, tn +ν )),

M [u (x, tn); k] :=p ν=0 (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 Lis defined as the order of(, σ) for (1) It is easily verified

that, for all sufficiently smooth functions F m, the operatorsLandMare related by

where m = l, , N − p, also z∗(x, tn +ν ) lies between Z(x, tn +ν ) and z(x, tn +ν ).

Definition 2.2: LetL be of order p∗, and have order q∗ Then, we define the order r∗of the linear

(H2) Fm(x, t, u, z) − Fm(x, t, ˜u, z)∞≤ L1u − ˜u∞, ∀(x, t, u, z), (x, t, ˜u, z) ∈ H,

(H3) Fm(x, t, u, z) −Fm(x, t, ˜u, z)∞≤ L2u − ˜u∞, ∀(x, t, u, z), (x, t, ˜u, z) ∈ H,

for some Li ≥ 0, i = 1, , 5 Therefore, if

(i) F m in (1) satisfy the hypothesis H1, and H2,

(ii) Fm in (6) satisfy the hypothesis H1and H3,

(iii) the p-step method (, σ) is zero-stable,

(iv) the p-step method (, σ; ) is of at least order one.

Then,

(1) under the hypothesis H1and 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 =t n un(x) = u(x, t),

holds for all t ∈ [0, T], and for all solutions {u n(x)} of (3) satisfying starting conditions un(k) = un(x) for which limk→0un(k) = u0, n = 0, , p − 1, and J(ν, ν) for which lim k→0J(ν, ν) = 0, ν =

1, , l − 1 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 → 0 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]) (seefor 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

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, k2η)-plane is said to be a interval region of absolute

stability of the multi-step method, if for all(kξ, k2η) ∈  the method multi-step is absolutely stable.

Definition 2.4: The method (, σ; ) is said to be interval A-stable if its region  of absolute stability

contains the quarter plane k ξ < 0, k2η < 0

In practical applications, once the region for a particular method has been established, estimatesforξ 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, k2η) ⊆ .

3 Implementation of HGFDM

In this section, we illustrate the proposed method and give a starting procedure for its implementationfor both bounded and unbounded spatial domains in two sections

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 step methods to solve the parabolic Volterra integro-differential equations of the second kind (6) with

multi-∂ = {−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

of J−2 ordinary differential equation subject to the following initial-boundary conditions

Using these matrices, Equations (8) and (9) can be rewritten as

Ut(t) = A1U m (t) +1(t), t0, m= 1, 2,

whereA1[ ] := E[ ] and1(t) = Z(t) + G(t) (It is better to use the different notations in bounded

and unbounded domains, so we introduceA1for 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 = 1 then the solution of the problem (10) is given by

−A11 0 , Z (t) = [v(t)U(t), v(t)] T , Z0 = [v(0)U0, v (0)] T and v (t) =

v (0)e0t−A11U(ξ) dξ is the integration factor.

Proof: (i) Multiplying (10) by the integrating factor e−A11and integrate from 0 to t to get the desired

result Since

Ut (t)e−A11−d A11

dt U (t)e−A11=1(t)e−A11.Hence, we write

Now sinceA11[ ](0) = 0 therefore we get the desired results.

(ii) We represent the Riccati differential equation for (10) in the projective spaceZ = (vU, v) as a

system of two first-order linear ordinary differential equations [19]

(ii) Using (12), we derive the stability estimates if A11 ∞< α and 1(t)∞< β then

(ii) Second, by using (12) we have,

Remark 3.1: In the same way as for (10), it follows, if U and  U are solutions of (10) when m= 1with{U0, Z (t), G(t),k(t)} and { U0, Z(t),  G(t),  k(t)} then for (i) 4|t|

U(t) −  U (t)∞≤ (1 + U0− U0 ∞)e |t| max{α,β}

In Theorem 3.2, the stability estimates have the growth in the order of e4|t|h2 It seems that the estimate

is not sharp by noting that the spectral radius ofA1is negative Thus, it is hard to prove the decay, but

it is not necessary to have the growth in the order of e4|t|h2 The model problems studied here seems that

it holds the maximum principle Therefore, based on the above remark, we conclude the followingcorollary

Corollary 3.2: For m = 1, (10) is conditionally well posedness for4|t|h2 = 0 when

Finally, we observe that when m = 2 and max{α, β} is very small near zero then (10) is

condition-ally well posedness In the next section, we choose a class of stable schemes

Wn:= U(t n), Fm,n:=Fm,n(tn , U (tn), Z(tn)),

for n = 0, , N Hence, we consider W N (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 AM4method 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 W N (t) is the approximated

solution of problem (10) applying RK4 and AM4 methods

Corollary 3.3: Let U(t) and W N(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), δk(t)} then for (i) m = 1, α = 4|t|

For (iii) m = 2 and T is very small near zero, we have

U(t) − W N(t)∞≤ (1 + δU0 ∞)emax{α,β}.Finally for (i), (ii) and (iii) we have

lim

N→∞U(t) − W N (t)∞= 0

Proof: From Theorem 2.1, Corollary 3.1 and Remark 3.1, lim N→∞δU0 ∞= 0, limN→∞

δk(t)∞= 0 and limN→∞δG(t)∞= 0, we immediately conclude the assertion