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

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

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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

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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.

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

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

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

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

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

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Table 1. Matrix representation of coefficients of GLM.

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

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

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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 $

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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 $

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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 $

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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 $

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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} $

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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} $

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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} $

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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} $

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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} $

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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 $

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

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