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Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind
Applied Mathematics and Computation 165 (2005) 223–227 www.elsevier.com/locate/amc
Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind E. Babolian, A. Davari
*
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Islamic Azad University of Qazvin, 599 Teleghani Avenue, Tehran 15618, Iran
Abstract In this paper, we propose new ideas to implement Adomian decomposition method to solve Volterra integral equations. Numerical examples are presented to illustrate the method for linear Volterra integral equations of the second kind. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction Many papers have been written [2,5,7] in which Adomian method is used for solving linear and nonlinear equations, especially integral equations [5,6]. Consider the following linear Volterra integral equation
*
Corresponding author. E-mail address: [email protected] (A. Davari).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.065
224
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
uðxÞ ¼ f ðxÞ þ
Z
x
x 2 ½0; 1:
Kðx; tÞuðtÞ dt
ð1Þ
0
Adomian process assumes uðxÞ ¼ tain um(x) recursively by
P1
m¼0 um ðxÞ
and substitutes it in (1), to ob-
u0 ðxÞ ¼ f ðxÞ; Z x Kðx; tÞum ðtÞ dt: umþ1 ðxÞ ¼
ð2Þ
0
When the P1kernel of the integral equation is complicated or calculating terms of series m¼0 um ðxÞ are difficult or impossible analytically [1,3], Adomian method needs some modifications. In the following section we deal with these subjects.
2. Discussion An obvious numerical procedure is to approximate the integral term in (2) via a quadrature rule which integrals with respect to the variable t for a fixed value of x. It is natural to choose a regular mesh in x, thus setting x = xi = ih where h ¼ m1 is the fixed step length. We would approximate in an obvious notation [3,4] the integral term in Eq. (2) by Z xi i i X X Kðxi ; tÞum ðtÞ dt ’ h wij Kðxi ; tj Þum ðtj Þ ¼ h wij K ij um ðtj Þ; 0
j¼0
j¼0
where xi = ti, i = 0, 1, . . . ,m. This leads to the following set of linear equations umþ1 ðti Þ ’ h
i X
wij Kðti ; tj Þum ðtj Þ;
i ¼ 0; 1; . . . m ¼ 0; 1; 2; . . . :
j¼0
For choosing suitable weights wij, we note that for each i the set wij, j = 0, 1, . . ., i represents the weights for an (i + 1)-points quadrature rules of Newton–Cotes type for the interval [0, ih]. We implement the above idea on the following example with m = 20 or h ¼ 201 . Tables 1–4 show P20 u20 ðxi Þ ¼ i¼1 uðxi Þ; uðxi Þ and errors at points xi, i = 0, 1, . . ., m. Example 1. Consider the following Volterra integral equation Z x x3 x4 uðxÞ ¼ x x2 þ þ þ ðt xÞuðtÞ dt; 0 < x < 1; 6 12 0 with the exact solution u(x) = x x2.
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
225
Table 1 The exact approximate value of u(x), and its errors for Example 1 x
Exact u(x)
Approximation u(x)
Errors
0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.047500000 0.090000000 0.160000000 0.210000000 0.240000000 0.250000000 0.240000000 0.210000000 0.160000000 0.090000000 0
0 0.04752031 0.08999993 0.16000024 0.21000184 0.24000542 0.25001140 0.24001998 0.21003110 0.16004446 0.09005950 0.00007543
0 0.0000203
0.6770E 6 0.24373E 5 0.1847876E 5 0.542314E 5 0.00001140 0.00001998 0.00003110 0.00004446 0.00005950 0.00007543
Table 2 The exact approximate values of u(x), and its errors for Example 2 x
Exact u(x)
Approximation u(x)
Errors
0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.04997917 0.09983342 0.19866933 0.29552021 0.38941834 0.47942554 0.56464247 0.64421769 0.71735609 0.78332691 0.84147098
0 0.05000000 0.09983333 0.19866958 0.29552231 0.38942488 0.47944013 0.56466968 0.64426292 0.71742550 0.78342727 0.84160954
0 0.00002083
0.833134E 7 0.250454E 6 0.210334E 5 0.654704E 5 0.000014601 0.000027212 0.000045236 0.000069415 0.000100366 0.000138562
Example 2 uðxÞ ¼ x þ
Z
x
ðt xÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = sin x.
Example 3 uðxÞ ¼ 1 þ
Z
x
ðx tÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = cosh x.
226
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
Table 3 The exact approximate values of u(x), and its errors for Example 3 x
Exact u(x)
Approximation u(x)
Error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1.0050042 1.0200668 1.0453385 1.0810724 1.1276260 1.1854652 1.2551690 1.3374350 1.4330864 1.5430806
1 1.00500972 1.02008904 1.04538899 1.08116298 1.12776937 1.18567506 1.25546016 1.33782378 1.43359109 1.54372151
0 0.55541E 5 0.00002229 0.00005048 0.00009061 0.00014341 0.00020985 0.00029115 0.00038884 0.00050470 0.00064087
Table 4 The exact approximate values of u(x), and its errors for Example 4 x
Exact u(x)
Approximation u(x)
Error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 0.899833250 0.798663998 0.695479707 0.589247674 0.478904694 0.363346418 0.241416298 0.111894018
0.02651673
0.17520119
1 0.89983888 0.79868602 0.69552792 0.58933100 0.47903126 0.36352357 0.24165066 0.11219147
0.02615105
0.17476298
0 0.563890E-5 0.00002202 0.00004821 0.00008333 0.00012656 0.00017715 0.00023436 0.00029745 0.00036566 0.00043821
Example 4 x2 uðxÞ ¼ 1 x þ 2
Z
x
ðx tÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = 1 sinh x.
3. Conclusion Rx We know that the integral 0 f ðtÞ dt is not available in some cases [1] and practically the Adomain decomposition method fails. We illustrated that in this cases we can to use numerical integration and solve Volterra integral equations with Adomian method.
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
227
References [1] E. Babolian, A. Davari, Numerical implementation of Adomain decomposition method, Applied mathematics and computation 153 (2004) 301–305. [2] Y. Cherruault, G. Adomian, K. Abbaoui, R. Rach, Further remarks on convergence of decomposition method, International Journal of Biomedical Computing 38 (1995) 89–93. [3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, 1985. [4] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 2002. [5] A.M. Wazwaz, S.A. Khuri, Two methods for solving integral equations, Applied mathematics and computation 77 (1996) 79–89. [6] A.M. Wazwaz, A First Course in Integral Equations, WSPC, New Jersey, 1997. [7] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied mathematics and computation III (2000) 53–69.
Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind E. Babolian, A. Davari
*
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Islamic Azad University of Qazvin, 599 Teleghani Avenue, Tehran 15618, Iran
Abstract In this paper, we propose new ideas to implement Adomian decomposition method to solve Volterra integral equations. Numerical examples are presented to illustrate the method for linear Volterra integral equations of the second kind. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction Many papers have been written [2,5,7] in which Adomian method is used for solving linear and nonlinear equations, especially integral equations [5,6]. Consider the following linear Volterra integral equation
*
Corresponding author. E-mail address: [email protected] (A. Davari).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.065
224
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
uðxÞ ¼ f ðxÞ þ
Z
x
x 2 ½0; 1:
Kðx; tÞuðtÞ dt
ð1Þ
0
Adomian process assumes uðxÞ ¼ tain um(x) recursively by
P1
m¼0 um ðxÞ
and substitutes it in (1), to ob-
u0 ðxÞ ¼ f ðxÞ; Z x Kðx; tÞum ðtÞ dt: umþ1 ðxÞ ¼
ð2Þ
0
When the P1kernel of the integral equation is complicated or calculating terms of series m¼0 um ðxÞ are difficult or impossible analytically [1,3], Adomian method needs some modifications. In the following section we deal with these subjects.
2. Discussion An obvious numerical procedure is to approximate the integral term in (2) via a quadrature rule which integrals with respect to the variable t for a fixed value of x. It is natural to choose a regular mesh in x, thus setting x = xi = ih where h ¼ m1 is the fixed step length. We would approximate in an obvious notation [3,4] the integral term in Eq. (2) by Z xi i i X X Kðxi ; tÞum ðtÞ dt ’ h wij Kðxi ; tj Þum ðtj Þ ¼ h wij K ij um ðtj Þ; 0
j¼0
j¼0
where xi = ti, i = 0, 1, . . . ,m. This leads to the following set of linear equations umþ1 ðti Þ ’ h
i X
wij Kðti ; tj Þum ðtj Þ;
i ¼ 0; 1; . . . m ¼ 0; 1; 2; . . . :
j¼0
For choosing suitable weights wij, we note that for each i the set wij, j = 0, 1, . . ., i represents the weights for an (i + 1)-points quadrature rules of Newton–Cotes type for the interval [0, ih]. We implement the above idea on the following example with m = 20 or h ¼ 201 . Tables 1–4 show P20 u20 ðxi Þ ¼ i¼1 uðxi Þ; uðxi Þ and errors at points xi, i = 0, 1, . . ., m. Example 1. Consider the following Volterra integral equation Z x x3 x4 uðxÞ ¼ x x2 þ þ þ ðt xÞuðtÞ dt; 0 < x < 1; 6 12 0 with the exact solution u(x) = x x2.
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
225
Table 1 The exact approximate value of u(x), and its errors for Example 1 x
Exact u(x)
Approximation u(x)
Errors
0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.047500000 0.090000000 0.160000000 0.210000000 0.240000000 0.250000000 0.240000000 0.210000000 0.160000000 0.090000000 0
0 0.04752031 0.08999993 0.16000024 0.21000184 0.24000542 0.25001140 0.24001998 0.21003110 0.16004446 0.09005950 0.00007543
0 0.0000203
0.6770E 6 0.24373E 5 0.1847876E 5 0.542314E 5 0.00001140 0.00001998 0.00003110 0.00004446 0.00005950 0.00007543
Table 2 The exact approximate values of u(x), and its errors for Example 2 x
Exact u(x)
Approximation u(x)
Errors
0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.04997917 0.09983342 0.19866933 0.29552021 0.38941834 0.47942554 0.56464247 0.64421769 0.71735609 0.78332691 0.84147098
0 0.05000000 0.09983333 0.19866958 0.29552231 0.38942488 0.47944013 0.56466968 0.64426292 0.71742550 0.78342727 0.84160954
0 0.00002083
0.833134E 7 0.250454E 6 0.210334E 5 0.654704E 5 0.000014601 0.000027212 0.000045236 0.000069415 0.000100366 0.000138562
Example 2 uðxÞ ¼ x þ
Z
x
ðt xÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = sin x.
Example 3 uðxÞ ¼ 1 þ
Z
x
ðx tÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = cosh x.
226
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
Table 3 The exact approximate values of u(x), and its errors for Example 3 x
Exact u(x)
Approximation u(x)
Error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1.0050042 1.0200668 1.0453385 1.0810724 1.1276260 1.1854652 1.2551690 1.3374350 1.4330864 1.5430806
1 1.00500972 1.02008904 1.04538899 1.08116298 1.12776937 1.18567506 1.25546016 1.33782378 1.43359109 1.54372151
0 0.55541E 5 0.00002229 0.00005048 0.00009061 0.00014341 0.00020985 0.00029115 0.00038884 0.00050470 0.00064087
Table 4 The exact approximate values of u(x), and its errors for Example 4 x
Exact u(x)
Approximation u(x)
Error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 0.899833250 0.798663998 0.695479707 0.589247674 0.478904694 0.363346418 0.241416298 0.111894018
0.02651673
0.17520119
1 0.89983888 0.79868602 0.69552792 0.58933100 0.47903126 0.36352357 0.24165066 0.11219147
0.02615105
0.17476298
0 0.563890E-5 0.00002202 0.00004821 0.00008333 0.00012656 0.00017715 0.00023436 0.00029745 0.00036566 0.00043821
Example 4 x2 uðxÞ ¼ 1 x þ 2
Z
x
ðx tÞuðtÞ dt;
0 < x < 1;
0
with the exact solution u(x) = 1 sinh x.
3. Conclusion Rx We know that the integral 0 f ðtÞ dt is not available in some cases [1] and practically the Adomain decomposition method fails. We illustrated that in this cases we can to use numerical integration and solve Volterra integral equations with Adomian method.
E. Babolian, A. Davari / Appl. Math. Comput. 165 (2005) 223–227
227
References [1] E. Babolian, A. Davari, Numerical implementation of Adomain decomposition method, Applied mathematics and computation 153 (2004) 301–305. [2] Y. Cherruault, G. Adomian, K. Abbaoui, R. Rach, Further remarks on convergence of decomposition method, International Journal of Biomedical Computing 38 (1995) 89–93. [3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, 1985. [4] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 2002. [5] A.M. Wazwaz, S.A. Khuri, Two methods for solving integral equations, Applied mathematics and computation 77 (1996) 79–89. [6] A.M. Wazwaz, A First Course in Integral Equations, WSPC, New Jersey, 1997. [7] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied mathematics and computation III (2000) 53–69.