Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method
- Atefeh Fahim^{1},
- Mohammad Ali Fariborzi Araghi^{1}Email authorView ORCID ID profile,
- Jalil Rashidinia^{1, 2} and
- Mehdi Jalalvand^{3}
https://doi.org/10.1186/s13662-017-1416-7
© The Author(s) 2017
Received: 10 July 2017
Accepted: 31 October 2017
Published: 10 November 2017
Abstract
We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an important case in this study. The convergence analysis has been discussed in detail, which shows that the approach exponentially converges to the solution. Furthermore, numerical examples and illustrations are presented to prove the validity of the suggested method.
Keywords
MSC
1 Introduction
Modeling phenomena in viscoelasticity, biological models, chemical kinetics, heat conduction in materials with memory, population dynamics, fluid dynamics and nuclear reactor dynamics, mathematical biology, financial mathematics, compression of viscoelastic media, and other similar areas are all done by partial integro-differential equations of type (1). See, for example, [5] and the references therein. This problem governs many physical systems occurring in diffusion problems as a particular case [6].
To treat the partial integro-differential equations (PIDEs), a substantial number of methods have been applied. For example, the pseudo-spectral Legendre-Galerkin method for solving a parabolic PIDE with convolution-type kernel was presented in [7]. Combination of radial basis functions and finite difference for solving nonlinear-type PIDEs with smooth kernel containing an unknown function was considered in [8]. Also, a spectral method was proposed in [9] for the PIDEs with a weakly singular kernel.
The numerical solution of equation (1) with a weakly singular kernel was considered by many authors, such as finite-element methods [3, 10], finite-difference methods [11, 12], compact difference schemes [13], spectral collocation methods [14], orthogonal spline collocation methods [15], variational iteration and Adomian decomposition methods [16], radial basis functions methods [17], and quasi-wavelet methods [18]. However, construction of precise numerical methods for integro-differential equations is still a challenge owing to the weak singularity of the kernel \(k_{0}\) that contains sharp states of transitions in the solution. This lack of smoothness of the solution near \(t=0\) results in a decay in the order of the practical performance of familiar timestepping methods for equation (1). For instance, the trapezoidal rule with product integration of the quadrature term does not produce expected \(\mathcal{O}({\Delta t}^{2})\) errors [19].
The sinc approximation has been studied by many authors to solve various equations such as integral equations [20], ordinary differential equations [21], partial differential equations [22–24], integro-differential equations [25], and so on, due to high accuracy, exponential rate of convergence, and near optimality of this method [26]. With these backgrounds, we extend the sinc-collocation method for solving partial integro-differential equations of type (1).
In this paper, the time discretization method to solve equation (1) is effected by a combination of finite difference and quadrature. For this purpose, we apply the backward Euler method in addition to the product trapezoidal integration rule [19] for the integral term. Consequently, equation (1) is reduced to a system of ordinary differential equations (ODEs), which is discretized with the sinc-collocation method. In addition, the accuracy and efficiency of the suggested method is tested with some examples and illustrations.
This paper is organized as follows. Section 2 provides some basic definitions, assumptions, and preliminaries of sinc approximation. In Section 3, we develop the sinc collocation method to solve Volterra partial integro-differential equations. In Section 4, we discuss the convergence analysis of the proposed method. Finally, in Section 5, numerical examples are solved to verify the accuracy and efficiency of the proposed approach.
2 Preliminaries
The goal of this section is to recall notation and definitions of the sinc function and state some known theorems important for the rest of this paper, which were discussed thoroughly in [27, 28].
Definition 1
([27], p. 59)
Definition 2
([28], p. 180)
The following theorem presents the convergence result on the approximation of derivatives particularly useful for approximate solving some differential equations.
Theorem 1
([28], p. 208)
The sinc-collocation method requires the derivatives of the composite sinc function to be evaluated at the nodes. So, we need to recall the following lemma.
Lemma 1
([27], p. 106)
3 Description of the method
3.1 Discretization in time
3.2 Discretization in space: sinc-collocation method
4 Convergence analysis
We need to derive an upper bound for \(\Vert P^{-1}\Vert_{2}\), which is given in the following lemma.
Lemma 2
Proof
The following theorem gives a bound for \(\vert u_{m}^{n+1}(x)-w^{n+1}_{m}(x)\vert\).
Theorem 2
Proof
Theorem 3
Proof
Remark
5 Numerical results
Example 1
Comparison of estimated maximum pointwise errors of Example 1 for \(\pmb{N = 4}\) , \(\pmb{T=1}\) at \(\pmb{t=1}\) and different values of x
x | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | ||
Δt = 0.01 | SMLP | 1.9 × 10^{−3} | 3.4 × 10^{−3} | 4.4 × 10^{−3} | 5.1 × 10^{−3} | 5.3 × 10^{−3} | 5.1 × 10^{−3} | 4.4 × 10^{−3} | 3.4 × 10^{−3} | 1.9 × 10^{−3} |
SMTR | 7.6 × 10^{−4} | 8.7 × 10^{−4} | 8.9 × 10^{−4} | 9.1 × 10^{−4} | 9.3 × 10^{−4} | 9.2 × 10^{−4} | 8.6 × 10^{−4} | 7.5 × 10^{−4} | 6.7 × 10^{−4} | |
Δt = 0.001 | SMLP | 4.1 × 10^{−4} | 6.1 × 10^{−4} | 5.3 × 10^{−4} | 4.9 × 10^{−4} | 4.9 × 10^{−4} | 4.9 × 10^{−4} | 5.3 × 10^{−4} | 6.1 × 10^{−4} | 4.1 × 10^{−4} |
SMTR | 1.2 × 10^{−4} | 1.8 × 10^{−4} | 1.4 × 10^{−4} | 2.1 × 10^{−4} | 9.2 × 10^{−5} | 9.6 × 10^{−5} | 8.9 × 10^{−5} | 1.6 × 10^{−4} | 2.3 × 10^{−4} | |
Δt = 0.0001 | TPEM | 7.5 × 10^{−3} | 7.5 × 10^{−3} | 7.6 × 10^{−3} | 7.4 × 10^{−3} | 7.5 × 10^{−3} | 7.4 × 10^{−3} | 7.3 × 10^{−3} | 7.7 × 10^{−3} | 7.8 × 10^{−3} |
TPIM | 7.1 × 10^{−3} | 7.2 × 10^{−3} | 7.4 × 10^{−3} | 7.5 × 10^{−3} | 7.5 × 10^{−3} | 7.3 × 10^{−3} | 7.2 × 10^{−3} | 7.4 × 10^{−3} | 7.6 × 10^{−3} | |
CNM | 5.1 × 10^{−3} | 5.2 × 10^{−3} | 5.3 × 10^{−3} | 5.2 × 10^{−3} | 5.4 × 10^{−3} | 5.3 × 10^{−3} | 5.5 × 10^{−3} | 5.3 × 10^{−3} | 5.2 × 10^{−3} | |
CM | 6.2 × 10^{−4} | 6.1 × 10^{−4} | 6.5 × 10^{−4} | 6.6 × 10^{−5} | 6.6 × 10^{−5} | 6.6 × 10^{−5} | 6.4 × 10^{−4} | 6.5 × 10^{−4} | 6.4 × 10^{−4} | |
SMLP | 2.6 × 10^{−4} | 3.1 × 10^{−4} | 1.3 × 10^{−4} | 1.1 × 10^{−5} | 1.7 × 10^{−5} | 1.1 × 10^{−5} | 1.3 × 10^{−4} | 3.1 × 10^{−4} | 2.6 × 10^{−4} | |
SMTR | 7.5 × 10^{−5} | 8.1 × 10^{−5} | 5.7 × 10^{−5} | 9.2 × 10^{−6} | 9.5 × 10^{−6} | 9.1 × 10^{−6} | 7.8 × 10^{−5} | 8.6 × 10^{−5} | 8.2 × 10^{−5} |
Results for Example 1 at \(\pmb{t=0.01}\)
N | SMLP | SMTR | \(\boldsymbol {\operatorname{Cond}(P)= \Vert P \Vert \Vert {{P^{ - 1}}} \Vert }\) |
---|---|---|---|
4 | 1.10 × 10^{−2} | 8.27 × 10^{−3} | 4.61 × 10^{2} |
8 | 2.85 × 10^{−3} | 7.94 × 10^{−4} | 6.71 × 10^{3} |
16 | 4.68 × 10^{−4} | 9.32 × 10^{−5} | 3.88 × 10^{4} |
32 | 9.75 × 10^{−5} | 4.71 × 10^{−5} | 1.30 × 10^{5} |
Example 2
Results for Example 2
n | \(\boldsymbol{\Delta t = 10^{-5}}\) | \(\boldsymbol{\Delta t = 10^{-6}}\) | ||||
---|---|---|---|---|---|---|
QWM | SMLP | SMTR | QWM | SMLP | SMTR | |
50 | 4.9343e − 004 | 5.1621e − 005 | 7.5561e − 006 | 1.5630e − 005 | 9.8142e − 006 | 2.7356e − 006 |
150 | 2.5228e − 003 | 2.9006e − 004 | 8.6902e − 005 | 8.0470e − 005 | 9.8142e − 006 | 2.9441e − 006 |
250 | 5.3616e − 003 | 6.4177e − 004 | 1.1924e − 004 | 1.7272e − 004 | 2.0303e − 005 | 9.1242e − 006 |
350 | 8.7631e − 003 | 1.0796e − 003 | 6.6327e − 004 | 2.8572e − 004 | 3.4165e − 005 | 1.5836e − 005 |
450 | 1.2588e − 002 | 1.5898e − 003 | 1.1745e − 004 | 4.1611e − 004 | 5.0328e − 005 | 2.3042e − 005 |
Example 3
Results for Example 3
N | \(\boldsymbol{\Delta t = 10^{-5}}\) | \(\boldsymbol{\Delta t =10^{-6}}\) | \(\boldsymbol{\Delta t =10^{-7}}\) | |||
---|---|---|---|---|---|---|
SMLP | SMTR | SMLP | SMTR | SMLP | SMTR | |
8 | 2.75 × 10^{−3} | 8.52 × 10^{−4} | 1.05 × 10^{−3} | 8.28 × 10^{−4} | 3.57 × 10^{−6} | 9.81 × 10^{−7} |
2.83 × 10^{−3} | 8.94 × 10^{−4} | 2.58 × 10^{−3} | 8.41 × 10^{−4} | 1.65 × 10^{−4} | 7.34 × 10^{−6} | |
2.87 × 10^{−3} | 9.86 × 10^{−4} | 2.61 × 10^{−3} | 9.73 × 10^{−4} | 4.35 × 10^{−4} | 6.52 × 10^{−5} | |
2.88 × 10^{−3} | 9.71 × 10^{−3} | 2.61 × 10^{−3} | 8.84 × 10^{−3} | 6.13 × 10^{−4} | 1.41 × 10^{−4} | |
16 | 2.75 × 10^{−4} | 7.32 × 10^{−5} | 2.63 × 10^{−4} | 7.28 × 10^{−5} | 9.00 × 10^{−5} | 6.87 × 10^{−7} |
2.92 × 10^{−4} | 6.54 × 10^{−5} | 2.66 × 10^{−4} | 1.79 × 10^{−5} | 2.56 × 10^{−4} | 5.62 × 10^{−6} | |
6.45 × 10^{−4} | 8.30 × 10^{−5} | 2.67 × 10^{−4} | 5.48 × 10^{−5} | 2.61 × 10^{−4} | 6.42 × 10^{−6} | |
1.08 × 10^{−3} | 9.24 × 10^{−5} | 2.67 × 10^{−4} | 6.74 × 10^{−5} | 2.61 × 10^{−4} | 7.83 × 10^{−6} | |
32 | 5.28 × 10^{−5} | 9.12 × 10^{−6} | 9.81 × 10^{−6} | 5.41 × 10^{−6} | 8.66 × 10^{−6} | 1.19 × 10^{−7} |
2.92 × 10^{−4} | 4.37 × 10^{−5} | 9.81 × 10^{−6} | 4.09 × 10^{−6} | 8.66 × 10^{−6} | 4.72 × 10^{−7} | |
6.45 × 10^{−4} | 5.83 × 10^{−5} | 2.04 × 10^{−5} | 6.49 × 10^{−6} | 8.66 × 10^{−6} | 7.28 × 10^{−7} | |
1.08 × 10^{−3} | 7.34 × 10^{−5} | 3.43 × 10^{−5} | 8.51 × 10^{−6} | 8.66 × 10^{−6} | 8.63 × 10^{−7} | |
64 | 5.28 × 10^{−5} | 3.41 × 10^{−6} | 1.67 × 10^{−6} | 9.86 × 10^{−7} | 6.11 × 10^{−8} | 7.53 × 10^{−8} |
2.92 × 10^{−4} | 1.92 × 10^{−5} | 9.24 × 10^{−6} | 5.17 × 10^{−6} | 2.28 × 10^{−7} | 8.91 × 10^{−8} | |
6.45 × 10^{−4} | 4.56 × 10^{−5} | 2.04 × 10^{−5} | 6.84 × 10^{−6} | 6.45 × 10^{−7} | 9.52 × 10^{−8} | |
1.08 × 10^{−3} | 5.37 × 10^{−5} | 3.43 × 10^{−5} | 7.13 × 10^{−6} | 9.44 × 10^{−7} | 1.54 × 10^{−7} |
6 Conclusions
In this paper, the sinc-collocation method was applied to solve linear Volterra partial integro-differential equations by using the Linsolve package and Tikhonov regularization methods for a final ill-conditioned system. To illustrate the effectiveness of the method, some examples were solved based on the proposed algorithm. Also, the convergence of the method was given. The results show that the proposed method is practically reliable and consistent in comparison with other mentioned methods, and using the Tikhonov regularization method for solving the final ill-conditioned algebraic system, the rate of convergence improved.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive comments to improve the quality of this work.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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