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December  2019, 9(4): 433-444. doi: 10.3934/naco.2019042

## 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

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  August 2019

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.

Citation: Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042
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##### References:
Log maximum error versus number of functions evaluations for Problem 1
Log maximum error versus number of functions evaluations for Problem 2
Log maximum error versus number of functions evaluations for Problem 3
Log maximum error versus number of functions evaluations for Problem 4
Log maximum error versus number of functions evaluations for Problem 5
Matrix representation of coefficients of GLM.
 $A_{s\times s}$ $U_{s\times r}$ $B_{r\times s}$ $V_{r\times r}$
 $A_{s\times s}$ $U_{s\times r}$ $B_{r\times s}$ $V_{r\times r}$
Matrix coefficients of GLM with $s = 3$, $r = 2$.
 $\left[ {\begin{array}{*{20}{l}} 0&0&0\\ {{a_{21}}}&0&0\\ {{a_{31}}}&{{a_{32}}}&0 \end{array}} \right]$ $\left[ {\begin{array}{*{20}{l}} {{u_{11}}}&{{u_{12}}}\\ {{u_{21}}}&{{u_{22}}}\\ {{u_{31}}}&{{u_{32}}} \end{array}} \right]$ $\left[ {\begin{array}{*{20}{l}} {{b_{11}}}&{{b_{12}}}&{{b_{13}}}\\ {{b_{21}}}&{{b_{22}}}&{{b_{23}}} \end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}} 1&{{v_{12}}}\\ {{v_{21}}}&0 \end{array}} \right]$
 $\left[ {\begin{array}{*{20}{l}} 0&0&0\\ {{a_{21}}}&0&0\\ {{a_{31}}}&{{a_{32}}}&0 \end{array}} \right]$ $\left[ {\begin{array}{*{20}{l}} {{u_{11}}}&{{u_{12}}}\\ {{u_{21}}}&{{u_{22}}}\\ {{u_{31}}}&{{u_{32}}} \end{array}} \right]$ $\left[ {\begin{array}{*{20}{l}} {{b_{11}}}&{{b_{12}}}&{{b_{13}}}\\ {{b_{21}}}&{{b_{22}}}&{{b_{23}}} \end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}} 1&{{v_{12}}}\\ {{v_{21}}}&0 \end{array}} \right]$
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 $b_{21}u_{12}\xi_{24} + b_{22}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{23}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12})$ $\qquad+u_{32}\xi_{24}\big)=0$
 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 $b_{21}u_{12}\xi_{24} + b_{22}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{23}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12})$ $\qquad+u_{32}\xi_{24}\big)=0$
Coefficients Set 1 of third order GLM
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{13}{18}$ $u_{21}=\frac{7}{9}$ $u_{22}=\frac{2}{9}$ $a_{31}=\frac{-17}{9}$ $a_{32}=2$ $u_{31}=\frac{17}{9}$ $u_{32}=\frac{-8}{9}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{13}{18}$ $u_{21}=\frac{7}{9}$ $u_{22}=\frac{2}{9}$ $a_{31}=\frac{-17}{9}$ $a_{32}=2$ $u_{31}=\frac{17}{9}$ $u_{32}=\frac{-8}{9}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
Coefficients Set 2 of third order GLM
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{2}{3}$ $u_{21}=\frac{5}{6}$ $u_{22}=\frac{1}{6}$ $a_{31}=\frac{-5}{3}$ $a_{32}=2$ $u_{31}=\frac{5}{3}$ $u_{32}=\frac{-2}{3}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{2}{3}$ $u_{21}=\frac{5}{6}$ $u_{22}=\frac{1}{6}$ $a_{31}=\frac{-5}{3}$ $a_{32}=2$ $u_{31}=\frac{5}{3}$ $u_{32}=\frac{-2}{3}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
Coefficients Set 3 of third order GLM
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{5}{6}$ $u_{21}=\frac{2}{3}$ $u_{22}=\frac{1}{3}$ $a_{31}=\frac{-7}{3}$ $a_{32}=2$ $u_{31}=\frac{7}{3}$ $u_{32}=\frac{-4}{3}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
 $u_{11}=1$ $u_{12}=0$ $a_{21}=\frac{5}{6}$ $u_{21}=\frac{2}{3}$ $u_{22}=\frac{1}{3}$ $a_{31}=\frac{-7}{3}$ $a_{32}=2$ $u_{31}=\frac{7}{3}$ $u_{32}=\frac{-4}{3}$ $b_{11}=\frac{1}{6}$ $b_{12}=\frac{2}{3}$ $b_{13}=\frac{1}{6}$ $v_{11}=1$ $v_{12}=0$ $b_{21}=0$ $b_{22}=0$ $b_{23}=0$ $v_{21}=1$ $v_{22}=0$
Maximum global errors for Problem 1
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $1.2347\times 10^{-6}$ $4.7137\times 10^{-6}$ $h=0.025$ $9.9859\times 10^{-9}$ $7.1772\times 10^{-8}$ $h=0.01$ $5.6041\times 10^{-10}$ $4.6094\times 10^{-9}$ $h=0.005$ $6.7079\times 10^{-11}$ $5.7715\times 10^{-10}$ $h=0.001$ $5.1845\times 10^{-13}$ $4.6243\times 10^{-12}$
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $1.2347\times 10^{-6}$ $4.7137\times 10^{-6}$ $h=0.025$ $9.9859\times 10^{-9}$ $7.1772\times 10^{-8}$ $h=0.01$ $5.6041\times 10^{-10}$ $4.6094\times 10^{-9}$ $h=0.005$ $6.7079\times 10^{-11}$ $5.7715\times 10^{-10}$ $h=0.001$ $5.1845\times 10^{-13}$ $4.6243\times 10^{-12}$
Maximum global errors for Problem 2
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $2.4606\times 10^{-6}$ $6.9906\times 10^{-6}$ $h=0.025$ $1.6319\times 10^{-8}$ $1.0137\times 10^{-7}$ $h=0.01$ $8.3870\times 10^{-10}$ $6.4622\times 10^{-9}$ $h=0.005$ $9.7077\times 10^{-11}$ $8.0749\times 10^{-10}$ $h=0.001$ $7.2935\times 10^{-13}$ $6.4604\times 10^{-12}$
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $2.4606\times 10^{-6}$ $6.9906\times 10^{-6}$ $h=0.025$ $1.6319\times 10^{-8}$ $1.0137\times 10^{-7}$ $h=0.01$ $8.3870\times 10^{-10}$ $6.4622\times 10^{-9}$ $h=0.005$ $9.7077\times 10^{-11}$ $8.0749\times 10^{-10}$ $h=0.001$ $7.2935\times 10^{-13}$ $6.4604\times 10^{-12}$
Maximum global errors for Problem 3
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $3.9332\times 10^{-6}$ $4.8141\times 10^{-5}$ $h=0.025$ $1.4323\times 10^{-7}$ $1.4400\times 10^{-6}$ $h=0.01$ $1.0939\times 10^{-8}$ $1.0134\times 10^{-7}$ $h=0.005$ $1.4325\times 10^{-9}$ $1.3052\times 10^{-8}$ $h=0.001$ $1.1851\times 10^{-11}$ $1.0688\times 10^{-10}$
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $3.9332\times 10^{-6}$ $4.8141\times 10^{-5}$ $h=0.025$ $1.4323\times 10^{-7}$ $1.4400\times 10^{-6}$ $h=0.01$ $1.0939\times 10^{-8}$ $1.0134\times 10^{-7}$ $h=0.005$ $1.4325\times 10^{-9}$ $1.3052\times 10^{-8}$ $h=0.001$ $1.1851\times 10^{-11}$ $1.0688\times 10^{-10}$
Maximum global errors for Problem 4
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $4.5416\times 10^{-6}$ $1.4256\times 10^{-5}$ $h=0.025$ $1.7061\times 10^{-8}$ $2.5908\times 10^{-7}$ $h=0.01$ $1.4270\times 10^{-9}$ $1.7075\times 10^{-8}$ $h=0.005$ $2.1002\times 10^{-10}$ $2.1553\times 10^{-9}$ $h=0.001$ $1.8836\times 10^{-12}$ $1.7377\times 10^{-11}$
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $4.5416\times 10^{-6}$ $1.4256\times 10^{-5}$ $h=0.025$ $1.7061\times 10^{-8}$ $2.5908\times 10^{-7}$ $h=0.01$ $1.4270\times 10^{-9}$ $1.7075\times 10^{-8}$ $h=0.005$ $2.1002\times 10^{-10}$ $2.1553\times 10^{-9}$ $h=0.001$ $1.8836\times 10^{-12}$ $1.7377\times 10^{-11}$
Maximum global errors for Problem 5
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $6.3429\times 10^{-6}$ $3.3432\times 10^{-5}$ $h=0.025$ $4.4142\times 10^{-8}$ $6.3237\times 10^{-7}$ $h=0.01$ $4.0164\times 10^{-9}$ $4.1259\times 10^{-8}$ $h=0.005$ $5.4245\times 10^{-10}$ $5.1811\times 10^{-9}$ $h=0.001$ $4.5689\times 10^{-12}$ $4.1572\times 10^{-11}$
 GLM, $s=3$ RK, $s=3$ Step size MAXE $h=0.1$ $6.3429\times 10^{-6}$ $3.3432\times 10^{-5}$ $h=0.025$ $4.4142\times 10^{-8}$ $6.3237\times 10^{-7}$ $h=0.01$ $4.0164\times 10^{-9}$ $4.1259\times 10^{-8}$ $h=0.005$ $5.4245\times 10^{-10}$ $5.1811\times 10^{-9}$ $h=0.001$ $4.5689\times 10^{-12}$ $4.1572\times 10^{-11}$
Total number of function evaluations Problems 1 - 5
 GLM, $s=3$ RK, $s=3$ Step size TFE $h=0.1$ $34$ $34$ $h=0.025$ $124$ $124$ $h=0.01$ $304$ $304$ $h=0.005$ $604$ $604$ $h=0.001$ $3004$ $3004$
 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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