Numerical solutions of Volterra integro-differential equations using General Linear Method

Faranak Rabiei; Fatin Abd Hamid; Zanariah Abd Majid; Fudziah Ismail

doi:10.3934/naco.2019042

Article Contents

2019, Volume 9, Issue 4: 433-444. Doi: 10.3934/naco.2019042

This issue Previous Article Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model Next Article System of generalized mixed nonlinear ordered variational inclusions

Numerical solutions of Volterra integro-differential equations using General Linear Method

1.
School of Engineering, Monash University Malaysia, 47500 Bandar Sunway, Selangor, Malaysia
2.
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

^* Corresponding author: faranak.rabiei@monash.edu; faranak.rabiei@gmail.com
^* Corresponding author: faranak.rabiei@monash.edu; faranak.rabiei@gmail.com

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received: February 2018

Revised: June 2018

Published: November 2019

Abstract Full Text(HTML) Figure(5) / Table(12) Related Papers Cited by

Abstract

In this paper, a third order General Linear Method for finding the numerical solution of Volterra integro-differential equation is considered. The order conditions of the proposed method are derived based on techniques of B-series and 'rooted trees'. The integral operator in Volterra integro-differential equation approximated using Simpson's rule and Lagrange interpolation is discussed. To illustrate the efficiency of third order General Linear Method, we compare the method with a third order Runge-Kutta method.

Keywords:

Volterra integro-differential equations,
general linear method,
quadrature rule,
third order method.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

FullText(HTML)

Figure 1. Log maximum error versus number of functions evaluations for Problem 1

Download: Full-size image PowerPoint slide

Figure 2. Log maximum error versus number of functions evaluations for Problem 2

Download: Full-size image PowerPoint slide

Figure 3. Log maximum error versus number of functions evaluations for Problem 3

Download: Full-size image PowerPoint slide

Figure 4. Log maximum error versus number of functions evaluations for Problem 4

Download: Full-size image PowerPoint slide

Figure 5. Log maximum error versus number of functions evaluations for Problem 5

Download: Full-size image PowerPoint slide

Table 1. Matrix representation of coefficients of GLM.

$ A_{s\times s} $	$ U_{s\times r} $
$ B_{r\times s} $	$ V_{r\times r} $

| Show Table

DownLoad: CSV

Table 2. Matrix coefficients of GLM with $ s = 3 $, $ r = 2 $.

$\left[ {\begin{array}{*{20}{l}} 0&0&0\\ {{a_{21}}}&0&0\\ {{a_{31}}}&{{a_{32}}}&0 \end{array}} \right] $	$\left[ {\begin{array}{*{20}{l}} {{u_{11}}}&{{u_{12}}}\\ {{u_{21}}}&{{u_{22}}}\\ {{u_{31}}}&{{u_{32}}} \end{array}} \right]$
$\left[ {\begin{array}{*{20}{l}} {{b_{11}}}&{{b_{12}}}&{{b_{13}}}\\ {{b_{21}}}&{{b_{22}}}&{{b_{23}}} \end{array}} \right]$	$\left[ {\begin{array}{*{20}{c}} 1&{{v_{12}}}\\ {{v_{21}}}&0 \end{array}} \right]$

| Show Table

DownLoad: CSV

Table 3. Order conditions of GLM up to order three.

No	Order conditions
1	$ b_{11}(u_{11}+u_{12})+b_{12}(u_{21}+u_{22})+b_{13}(u_{31}+u_{32})-v_{12}=1 $
2	$ b_{21}(u_{11}+u_{12})+b_{22}(u_{21}+u_{22})+b_{23}(u_{31}+u_{32})=0 $
3	$ -b_{11}u_{12}+b_{12}(a_{21}(u_{11} + u_{12})-u_{22})+b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big) $
	$ \qquad+v_{12}\xi_{22}=\frac{1}{2} $
4	$ -b_{21}u_{12}+b_{22}(a_{21}(u_{11} + u_{12})-u_{22})+b_{23}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)=0 $
5	$ b_{11}u_{12}^{2}+b_{12}\big(a_{21}(u_{11}+u_{12})-u_{22}\big)^{2}+b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)^{2} $
	$ \qquad+v_{12}\xi_{23}=\frac{1}{3} $
6	$ b_{21}u_{12}^{2}+b_{22}\big(a_{21}(u_{11}+u_{12})-u_{22}\big)^{2}+b_{23}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)^{2}=0 $
7	$ b_{11}u_{12}\xi_{22}+b_{12}\big(\xi_{22}u_{22} - a_{21}u_{12}\big)+b_{13}\big(-a_{31}u_{12} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22}) $
	$ \qquad +u_{32}\xi_{22}\big)+v_{12}\xi_{24}=\frac{1}{6} $
8	$ b_{21}u_{12}\xi_{22}+b_{22}\big(\xi_{22}u_{22} - a_{21}u_{12}\big)+b_{23}\big(-a_{31}u_{12} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22}) $
	$ \qquad+u_{32}\xi_{22}\big)=0 $
9	$ -b_{11}u_{12}^{3}+b_{12}(a_{21}(u_{11} + u_{12})-u_{22})^{3}+b_{13}\big(a_{31}(u_{11} + u_{12}) +a_{32}(u_{21} + u_{22})-u_{32}\big)^{3} $
	$ \qquad+v_{12}\xi_{25}\frac{1}{4}=\frac{1}{4} $
10	$ -b_{21}u_{12}^{3}+b_{22}(a_{21}(u_{11} + u_{12})-u_{22})^{3}+b_{23}\big(a_{31}(u_{11} + u_{12}) +a_{32}(u_{21} + u_{22}) $
	$ \qquad-u_{32}\big)^{3}=0 $
11	$ -b_{11}u_{12}^{2}\xi_{22} + b_{12}(a_{21}(u_{11} + u_{12})-u_{22})(\xi_{22}u_{22}-a_{21}u_{12}) +b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32} $
	$ \qquad (u_{21}+u_{22})-u_{32}\big)\big(-a_{31}u_{12}+a_{32}(a_{21}(u_{11}+u_{12})-u_{22})+u_{32}\xi_{22}\big)+v_{12}\xi_{26} =\frac{1}{8} $
12	$ -b_{21}u_{12}^{2}\xi_{22} + b_{22}(a_{21}(u_{11} + u_{12})-u_{22})(\xi_{22}u_{22}-a_{21}u_{12}) +b_{23}\big(a_{31}(u_{11} + u_{12}) $
	$ \qquad+a_{32}(u_{21} + u_{22})-u_{32}\big)\big(-a_{31}u_{12}+a_{32}(a_{21}(u_{11}+u_{12})-u_{22})+u_{32}\xi_{22}\big)=0 $
13	$ b_{11}u_{12}\xi_{23} + b_{12}(a_{21}u_{12}^{2}+\xi_{23}u_{22}) + b_{13}\big(a_{31}u_{12}^{2} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22})^{2}+u_{32}\xi_{23}\big) $
	$ \qquad+v_{12}\xi_{27}=\frac{1}{12} $
14	$ b_{21}u_{12}\xi_{23} + b_{22}(a_{21}u_{12}^{2}+\xi_{23}u_{22}) + b_{23}\big(a_{31}u_{12}^{2} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22})^{2} $
	$ \qquad+u_{32}\xi_{23}\big)=0 $
15	$ b_{11}u_{12}\xi_{24} + b_{12}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{13}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12})+u_{32}\xi_{24}\big) $
	$ \qquad+v_{12}\xi_{28}=\frac{1}{24} $
16	$ b_{21}u_{12}\xi_{24} + b_{22}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{23}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12}) $
	$ \qquad+u_{32}\xi_{24}\big)=0 $

| Show Table

DownLoad: CSV

Table 4. Coefficients Set 1 of third order GLM

			$ u_{11}=1 $	$ u_{12}=0 $
$ a_{21}=\frac{13}{18} $			$ u_{21}=\frac{7}{9} $	$ u_{22}=\frac{2}{9} $
$ a_{31}=\frac{-17}{9} $	$ a_{32}=2 $		$ u_{31}=\frac{17}{9} $	$ u_{32}=\frac{-8}{9} $
$ b_{11}=\frac{1}{6} $	$ b_{12}=\frac{2}{3} $	$ b_{13}=\frac{1}{6} $	$ v_{11}=1 $	$ v_{12}=0 $
$ b_{21}=0 $	$ b_{22}=0 $	$ b_{23}=0 $	$ v_{21}=1 $	$ v_{22}=0 $

| Show Table

DownLoad: CSV

Table 5. Coefficients Set 2 of third order GLM

			$ u_{11}=1 $	$ u_{12}=0 $
$ a_{21}=\frac{2}{3} $			$ u_{21}=\frac{5}{6} $	$ u_{22}=\frac{1}{6} $
$ a_{31}=\frac{-5}{3} $	$ a_{32}=2 $		$ u_{31}=\frac{5}{3} $	$ u_{32}=\frac{-2}{3} $
$ b_{11}=\frac{1}{6} $	$ b_{12}=\frac{2}{3} $	$ b_{13}=\frac{1}{6} $	$ v_{11}=1 $	$ v_{12}=0 $
$ b_{21}=0 $	$ b_{22}=0 $	$ b_{23}=0 $	$ v_{21}=1 $	$ v_{22}=0 $

| Show Table

DownLoad: CSV

Table 6. Coefficients Set 3 of third order GLM

			$ u_{11}=1 $	$ u_{12}=0 $
$ a_{21}=\frac{5}{6} $			$ u_{21}=\frac{2}{3} $	$ u_{22}=\frac{1}{3} $
$ a_{31}=\frac{-7}{3} $	$ a_{32}=2 $		$ u_{31}=\frac{7}{3} $	$ u_{32}=\frac{-4}{3} $
$ b_{11}=\frac{1}{6} $	$ b_{12}=\frac{2}{3} $	$ b_{13}=\frac{1}{6} $	$ v_{11}=1 $	$ v_{12}=0 $
$ b_{21}=0 $	$ b_{22}=0 $	$ b_{23}=0 $	$ v_{21}=1 $	$ v_{22}=0 $

| Show Table

DownLoad: CSV

Table 7. Maximum global errors for Problem 1

	GLM, $ s=3 $	RK, $ s=3 $
Step size	MAXE
$ h=0.1 $	$ 1.2347\times 10^{-6} $	$ 4.7137\times 10^{-6} $
$ h=0.025 $	$ 9.9859\times 10^{-9} $	$ 7.1772\times 10^{-8} $
$ h=0.01 $	$ 5.6041\times 10^{-10} $	$ 4.6094\times 10^{-9} $
$ h=0.005 $	$ 6.7079\times 10^{-11} $	$ 5.7715\times 10^{-10} $
$ h=0.001 $	$ 5.1845\times 10^{-13} $	$ 4.6243\times 10^{-12} $

| Show Table

DownLoad: CSV

Table 8. Maximum global errors for Problem 2

	GLM, $ s=3 $	RK, $ s=3 $
Step size	MAXE
$ h=0.1 $	$ 2.4606\times 10^{-6} $	$ 6.9906\times 10^{-6} $
$ h=0.025 $	$ 1.6319\times 10^{-8} $	$ 1.0137\times 10^{-7} $
$ h=0.01 $	$ 8.3870\times 10^{-10} $	$ 6.4622\times 10^{-9} $
$ h=0.005 $	$ 9.7077\times 10^{-11} $	$ 8.0749\times 10^{-10} $
$ h=0.001 $	$ 7.2935\times 10^{-13} $	$ 6.4604\times 10^{-12} $

| Show Table

DownLoad: CSV

Table 9. Maximum global errors for Problem 3

	GLM, $ s=3 $	RK, $ s=3 $
Step size	MAXE
$ h=0.1 $	$ 3.9332\times 10^{-6} $	$ 4.8141\times 10^{-5} $
$ h=0.025 $	$ 1.4323\times 10^{-7} $	$ 1.4400\times 10^{-6} $
$ h=0.01 $	$ 1.0939\times 10^{-8} $	$ 1.0134\times 10^{-7} $
$ h=0.005 $	$ 1.4325\times 10^{-9} $	$ 1.3052\times 10^{-8} $
$ h=0.001 $	$ 1.1851\times 10^{-11} $	$ 1.0688\times 10^{-10} $

| Show Table

DownLoad: CSV

Table 10. Maximum global errors for Problem 4

	GLM, $ s=3 $	RK, $ s=3 $
Step size	MAXE
$ h=0.1 $	$ 4.5416\times 10^{-6} $	$ 1.4256\times 10^{-5} $
$ h=0.025 $	$ 1.7061\times 10^{-8} $	$ 2.5908\times 10^{-7} $
$ h=0.01 $	$ 1.4270\times 10^{-9} $	$ 1.7075\times 10^{-8} $
$ h=0.005 $	$ 2.1002\times 10^{-10} $	$ 2.1553\times 10^{-9} $
$ h=0.001 $	$ 1.8836\times 10^{-12} $	$ 1.7377\times 10^{-11} $

| Show Table

DownLoad: CSV

Table 11. Maximum global errors for Problem 5

	GLM, $ s=3 $	RK, $ s=3 $
Step size	MAXE
$ h=0.1 $	$ 6.3429\times 10^{-6} $	$ 3.3432\times 10^{-5} $
$ h=0.025 $	$ 4.4142\times 10^{-8} $	$ 6.3237\times 10^{-7} $
$ h=0.01 $	$ 4.0164\times 10^{-9} $	$ 4.1259\times 10^{-8} $
$ h=0.005 $	$ 5.4245\times 10^{-10} $	$ 5.1811\times 10^{-9} $
$ h=0.001 $	$ 4.5689\times 10^{-12} $	$ 4.1572\times 10^{-11} $

| Show Table

DownLoad: CSV

Table 12. Total number of function evaluations Problems 1 - 5

	GLM, $ s=3 $	RK, $ s=3 $
Step size	TFE
$ h=0.1 $	$ 34 $	$ 34 $
$ h=0.025 $	$ 124 $	$ 124 $
$ h=0.01 $	$ 304 $	$ 304 $
$ h=0.005 $	$ 604 $	$ 604 $
$ h=0.001 $	$ 3004 $	$ 3004 $

| Show Table

DownLoad: CSV

Article Metrics

HTML views(1295) PDF downloads(844) Cited by(0)

[1]	J. C. Butcher, General linear methods, Acta Numerica, 15 (2006), 157-256. doi: 10.1017/S0962492906220014.
[2]	J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons, Chichester, 2008. doi: 10.1002/9781119121534.
[3]	P. Chartier, E. Hairer and G. Vilmart, Algebraic structures of B-series, Foundations of Computational Mathematics, 10 (2010), 407-427. doi: 10.1007/s10208-010-9065-1.
[4]	J. R. Dormand, Numerical Methods for Differential Equations: A Computational Approach, CRC Press, Florida, 1992. doi: 10.1201/9781351075107.
[5]	A. Filiz, A fourth-order robust numerical method for integro-differential equations, Asian Journal of Fuzzy and Applied Mathematics, 1 (2013), 28-33.
[6]	A. Filiz, Numerical solution of linear volterra integro-differential equations using runge-kutta-felhberg method, Applied and Computational Mathematics, 1 (2014), 9-14.
[7]	A. Filiz, General linear methods for ordinary differential equations, Mathematics and Computers in Simulation, 79 (2009), 1834-1845. doi: 10.1016/j.matcom.2007.02.006.
[8]	P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
[9]	F. Rabiei, F. A. Hamid, M. M. Rashidi and F. Ismail, Numerical simulation of fuzzy differential equations using general linear method and B-series, Advances in Mechanical Engineering, 9 (2010), 1-16.
[10]	B. Raftari, Numerical solutions of the linear volterra integro-differential equations: Homotopy perturbation method and finite difference method, World Applied Sciences Journal, 9 (2010), 7-12.
[11]	A. M. Wazwaz, Linear and Nonlinear Integral Equations, Springer, Beijing, 2011. doi: 10.1007/978-3-642-21449-3.
[12]	M. Zarebnia, Sinc numerical solution for the Volterra integro-differential equation, Nonlinear Sci. Numer. Simulat., 15 (2010), 700-706. doi: 10.1016/j.cnsns.2009.04.021.

Numerical solutions of Volterra integro-differential equations using General Linear Method

Abstract

References

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Numerical solutions of Volterra integro-differential equations using General Linear Method

Abstract

References

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content