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Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives
Laboratory of pure and applied Mathematics, , University of M'sila, 28000 M'sila, Algeria |
$ \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*} $ |
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. |



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