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September  2022, 12(3): 551-568. doi: 10.3934/naco.2021021

Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives

Miloud Moussai

Laboratory of Pure and Applied Mathematics, University of M’sila, 28000 M’sila, Algeria

Received  January 2021 Revised  May 2021 Published  September 2022 Early access  June 2021

The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation
$ \begin{equation*} \sum\limits_{k = 1}^{m}F_{k}(x)D^{(k\alpha )}y(x)+\lambda \int_{0}^{x}K(x, t)D^{(\alpha )}y(t)dt = g(x)y^{2}(x)+h(x)y(x)+P(x). \end{equation*} $
In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.
Keywords: Nonlinear fractional type Volterra integro-differential equation, Bernstein polynomials, collocation method, Caputo, Fractional derivative.
Mathematics Subject Classification: Primary: 45J05, 65R20.
Citation: Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 551-568. doi: 10.3934/naco.2021021
References:
[1]

A. K. AL-Juburee, Approximate solution for linear fredhom integro-differential equation and integral equation by using bernstein polynomials method, Journal of The College of Basic Education, 15 (2010), 11-20. 

[2]

Y. Çenesiz, Y. Keskin and A. Kurnaz, The solution of the bagley–torvik equation with the generalized taylor collocation method, Journal of the Franklin Institute, 347 (2010), 452-466.  doi: 10.1016/j.jfranklin.2009.10.007.

[3]

O. R. Işik, M. Sezer and Z. Güney, Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Applied Mathematics and Computation, 217 (2011), 7009-7020.  doi: 10.1016/j.amc.2011.01.114.

[4]

A. M. Kareem, A new definition of fractional derivative and fractional integral, Kirkuk University Journal For Scientific Studies, 13 (2018), 304-323. 

[5]

A. Mahdy, Numerical studies for solving fractional integro-differential equations, Journal of Ocean Engineering and Science, 3 (2018), 127-132. 

[6]

L. C. Miloud Moussai, A computational method based on bernstein polynomials for solving freedholm integro-differential equations under mixed conditions, Journal of Mathematics and Statistics, 13 (2017), 30-37. 

[7]

R. Mittal and R. Nigam, Solution of fractional integro-differential equations by adomian decomposition method, Int. J. Appl. Math. Mech., 4 (2008), 87-94. 

[8]

S. Momani and N. Shawagfeh, Decomposition method for solving fractional riccati differential equations, Applied Mathematics and Computation, 182 (2006), 1083-1092.  doi: 10.1016/j.amc.2006.05.008.

[9]

Z. Odibat and S. Momani, Modified homotopy perturbation method: application to quadratic riccati differential equation of fractional order, Chaos, Solitons & Fractals, 36 (2008), 167-174.  doi: 10.1016/j.chaos.2006.06.041.

[10]

K. Parand and S. A. Kaviani, Application of the exact operational matrices based on the bernstein polynomials, Journal of Mathematics and Computer Science, 6 (2013), 36-59. 

[11]

A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Applied Mathematical Modelling, 38 (2014), 1365-1372.  doi: 10.1016/j.apm.2013.08.007.

[12]

V. K. Singh, R. K. Pandey and O. P. Singh, New stable numerical solutions of singular integral equations of abel type by using normalized bernstein polynomials, Applied Mathematical Sciences, 3 (2009), 241-255.  doi: 10.1017/s0004972700029002.

[13]

L. Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 227 (2014), 66-76.  doi: 10.1016/j.amc.2013.11.004.

[14]

S. Yalç inbaş, M. Sezer and H. H. Sorkun, Legendre polynomial solutions of high-order linear fredholm integro-differential equations, Applied Mathematics and Computation, 210 (2009), 334-349.  doi: 10.1016/j.amc.2008.12.090.

[15]

J. Zhao, J. Xiao and N. J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numerical Algorithms, 65 (2014), 723-743.  doi: 10.1007/s11075-013-9710-2.

show all references

References:
[1]

A. K. AL-Juburee, Approximate solution for linear fredhom integro-differential equation and integral equation by using bernstein polynomials method, Journal of The College of Basic Education, 15 (2010), 11-20. 

[2]

Y. Çenesiz, Y. Keskin and A. Kurnaz, The solution of the bagley–torvik equation with the generalized taylor collocation method, Journal of the Franklin Institute, 347 (2010), 452-466.  doi: 10.1016/j.jfranklin.2009.10.007.

[3]

O. R. Işik, M. Sezer and Z. Güney, Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Applied Mathematics and Computation, 217 (2011), 7009-7020.  doi: 10.1016/j.amc.2011.01.114.

[4]

A. M. Kareem, A new definition of fractional derivative and fractional integral, Kirkuk University Journal For Scientific Studies, 13 (2018), 304-323. 

[5]

A. Mahdy, Numerical studies for solving fractional integro-differential equations, Journal of Ocean Engineering and Science, 3 (2018), 127-132. 

[6]

L. C. Miloud Moussai, A computational method based on bernstein polynomials for solving freedholm integro-differential equations under mixed conditions, Journal of Mathematics and Statistics, 13 (2017), 30-37. 

[7]

R. Mittal and R. Nigam, Solution of fractional integro-differential equations by adomian decomposition method, Int. J. Appl. Math. Mech., 4 (2008), 87-94. 

[8]

S. Momani and N. Shawagfeh, Decomposition method for solving fractional riccati differential equations, Applied Mathematics and Computation, 182 (2006), 1083-1092.  doi: 10.1016/j.amc.2006.05.008.

[9]

Z. Odibat and S. Momani, Modified homotopy perturbation method: application to quadratic riccati differential equation of fractional order, Chaos, Solitons & Fractals, 36 (2008), 167-174.  doi: 10.1016/j.chaos.2006.06.041.

[10]

K. Parand and S. A. Kaviani, Application of the exact operational matrices based on the bernstein polynomials, Journal of Mathematics and Computer Science, 6 (2013), 36-59. 

[11]

A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Applied Mathematical Modelling, 38 (2014), 1365-1372.  doi: 10.1016/j.apm.2013.08.007.

[12]

V. K. Singh, R. K. Pandey and O. P. Singh, New stable numerical solutions of singular integral equations of abel type by using normalized bernstein polynomials, Applied Mathematical Sciences, 3 (2009), 241-255.  doi: 10.1017/s0004972700029002.

[13]

L. Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 227 (2014), 66-76.  doi: 10.1016/j.amc.2013.11.004.

[14]

S. Yalç inbaş, M. Sezer and H. H. Sorkun, Legendre polynomial solutions of high-order linear fredholm integro-differential equations, Applied Mathematics and Computation, 210 (2009), 334-349.  doi: 10.1016/j.amc.2008.12.090.

[15]

J. Zhao, J. Xiao and N. J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numerical Algorithms, 65 (2014), 723-743.  doi: 10.1007/s11075-013-9710-2.

Figure 1.  Exact and approximate solutions for ($ n=3 $) Example 5.1
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Figure 2.  Exact and approximate solution for ($ n=4 $) Example 5.2
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Figure 3.  Exact and approximate solutions for ($ n=5 $) Example 5.3
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Table 1.  Exact and approximate solutions and square error for ($ n = 3 $) Example 5.1
$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0.0 $ $ 0.0 $ $ 0.0 $
$ 0.1 $ $ 0.2518 $ $ 0.2001 $ $ 0.042224 \times 10^{-2} $
$ 0.2 $ $ 0.4637 $ $ 0.5103 $ $ 0.0217156 \times 10^{-1} $
$ 0.3 $ $ 0.6355 $ $ 0.6858 $ $ 0.0253009 \times 10^{-1} $
$ 0.4 $ $ 0.7673 $ $ 0.7898 $ $ 0.050625 \times 10^{-2} $
$ 0.5 $ $ 0.8591 $ $ 0.8479 $ $ 0.012544 \times 10^{-2} $
$ 0.6 $ $ 0.9109 $ $ 0.8940 $ $ 0.028561 \times 10^{-2} $
$ 0.7 $ $ 0.9227 $ $ 0.8790 $ $ 0.0190969 \times 10^{-1} $
$ 0.8 $ $ 0.8946 $ $ 0.8790 $ $ 0.024336 \times 10^{-1} $
$ 0.9 $ $ 0.8264 $ $ 0.8259 $ $ 0.025 \times 10^{-5} $
$ 1 $ $ 0.7182 $ $ 0.7603 $ $ 0.0167246 \times 10^{-1} $
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$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0.0 $ $ 0.0 $ $ 0.0 $
$ 0.1 $ $ 0.2518 $ $ 0.2001 $ $ 0.042224 \times 10^{-2} $
$ 0.2 $ $ 0.4637 $ $ 0.5103 $ $ 0.0217156 \times 10^{-1} $
$ 0.3 $ $ 0.6355 $ $ 0.6858 $ $ 0.0253009 \times 10^{-1} $
$ 0.4 $ $ 0.7673 $ $ 0.7898 $ $ 0.050625 \times 10^{-2} $
$ 0.5 $ $ 0.8591 $ $ 0.8479 $ $ 0.012544 \times 10^{-2} $
$ 0.6 $ $ 0.9109 $ $ 0.8940 $ $ 0.028561 \times 10^{-2} $
$ 0.7 $ $ 0.9227 $ $ 0.8790 $ $ 0.0190969 \times 10^{-1} $
$ 0.8 $ $ 0.8946 $ $ 0.8790 $ $ 0.024336 \times 10^{-1} $
$ 0.9 $ $ 0.8264 $ $ 0.8259 $ $ 0.025 \times 10^{-5} $
$ 1 $ $ 0.7182 $ $ 0.7603 $ $ 0.0167246 \times 10^{-1} $
Table 2.  Exact and approximate solutions and square errors for ($ n=4 $) Example 5.2
$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0.0 $ $ 0.0 $ $ 0.0 $
$ 0.1 $ $ 0.110 $ $ 0.162 $ $ 0.02703\times 10^{-1} $
$ 0.2 $ $ 0.240 $ $ 0.280 $ $ 0.01600\times 10^{-1} $
$ 0.3 $ $ 0.390 $ $ 0.419 $ $ 0.03481\times 10^{-2} $
$ 0.4 $ $ 0.560 $ $ 0.579 $ $ 0.0361\times 10^{-2} $
$ 0.5 $ $ 0.750 $ $ 0.762 $ $ 0.0144\times 10^{-2} $
$ 0.6 $ $ 0.960 $ $ 0.965 $ $ 0.025\times 10^{-3} $
$ 0.7 $ $ 1.190 $ $ 1.186 $ $ 0.016\times 10^{-3} $
$ 0.8 $ $ 1.440 $ $ 1.426 $ $ 0.0196\times 10^{-3} $
$ 0.9 $ $ 1.710 $ $ 1.683 $ $ 0.0728\times 10^{-1} $
$ 1 $ $ 2.000 $ $ 1.957 $ $ 0.01849\times 10^{-1} $
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$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0.0 $ $ 0.0 $ $ 0.0 $
$ 0.1 $ $ 0.110 $ $ 0.162 $ $ 0.02703\times 10^{-1} $
$ 0.2 $ $ 0.240 $ $ 0.280 $ $ 0.01600\times 10^{-1} $
$ 0.3 $ $ 0.390 $ $ 0.419 $ $ 0.03481\times 10^{-2} $
$ 0.4 $ $ 0.560 $ $ 0.579 $ $ 0.0361\times 10^{-2} $
$ 0.5 $ $ 0.750 $ $ 0.762 $ $ 0.0144\times 10^{-2} $
$ 0.6 $ $ 0.960 $ $ 0.965 $ $ 0.025\times 10^{-3} $
$ 0.7 $ $ 1.190 $ $ 1.186 $ $ 0.016\times 10^{-3} $
$ 0.8 $ $ 1.440 $ $ 1.426 $ $ 0.0196\times 10^{-3} $
$ 0.9 $ $ 1.710 $ $ 1.683 $ $ 0.0728\times 10^{-1} $
$ 1 $ $ 2.000 $ $ 1.957 $ $ 0.01849\times 10^{-1} $
Table 3.  Exact and approximate solutions and square errors for ($ n=5 $) Example 5.3
$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0 $ $ 0 $ $ 0 $
$ 0.1 $ $ 0.316 $ $ 0.371 $ $ 0.01\times10^{-6} $
$ 0.2 $ $ 0.0594 $ $ 0.0897 $ $ 0.09193\times10^{-5} $
$ 0.3 $ $ 0.1643 $ $ 0.164465 $ $ 0.01225\times10^{-5} $
$ 0.4 $ $ 0.2530 $ $ 0.25336 $ $ 0.01296\times10^{-5} $
$ 0.5 $ $ 0.3535 $ $ 0.35402 $ $ 0.02704\times10^{-5} $
$ 0.6 $ $ 0.4646 $ $ 0.046525 $ $ 0.04225\times10^{-5} $
$ 0.7 $ $ 0.5857 $ $ 0.58621 $ $ 0.02601\times10^{-5} $
$ 0.8 $ $ 0.7155 $ $ 0.71527 $ $ 0.00525\times10^{-5} $
$ 0.9 $ $ 0.8538 $ $ 0.55476 $ $ 0.09216\times10^{-5} $
$ 1 $ $ 1 $ $ 1.000684 $ $ 0.00467856\times10^{-2} $
Table Options

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$ x_{i} $ $ exact \ solutions $ $ approximation \ solutions $ $ errors $
$ 0.0 $ $ 0 $ $ 0 $ $ 0 $
$ 0.1 $ $ 0.316 $ $ 0.371 $ $ 0.01\times10^{-6} $
$ 0.2 $ $ 0.0594 $ $ 0.0897 $ $ 0.09193\times10^{-5} $
$ 0.3 $ $ 0.1643 $ $ 0.164465 $ $ 0.01225\times10^{-5} $
$ 0.4 $ $ 0.2530 $ $ 0.25336 $ $ 0.01296\times10^{-5} $
$ 0.5 $ $ 0.3535 $ $ 0.35402 $ $ 0.02704\times10^{-5} $
$ 0.6 $ $ 0.4646 $ $ 0.046525 $ $ 0.04225\times10^{-5} $
$ 0.7 $ $ 0.5857 $ $ 0.58621 $ $ 0.02601\times10^{-5} $
$ 0.8 $ $ 0.7155 $ $ 0.71527 $ $ 0.00525\times10^{-5} $
$ 0.9 $ $ 0.8538 $ $ 0.55476 $ $ 0.09216\times10^{-5} $
$ 1 $ $ 1 $ $ 1.000684 $ $ 0.00467856\times10^{-2} $
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