
-
Previous Article
A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration
- NACO Home
- This Issue
-
Next Article
Convex optimization without convexity of constraints on non-necessarily convex sets and its applications in customer satisfaction in automotive industry
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. |



[1] |
Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030 |
[2] |
Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 |
[3] |
Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064 |
[4] |
Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053 |
[5] |
Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016 |
[6] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023 |
[7] |
Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 |
[8] |
Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044 |
[9] |
Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022028 |
[10] |
Yuling Guo, Zhongqing Wang. A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022052 |
[11] |
Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 |
[12] |
Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042 |
[13] |
Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 339-357. doi: 10.3934/dcdss.2021025 |
[14] |
Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223 |
[15] |
Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 621-636. doi: 10.3934/naco.2021026 |
[16] |
Piotr Grabowski. On analytic semigroup generators involving Caputo fractional derivative. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022014 |
[17] |
Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 |
[18] |
Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 |
[19] |
Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100 |
[20] |
Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4231-4253. doi: 10.3934/dcdsb.2021225 |
Impact Factor:
Tools
Article outline
Figures and Tables
[Back to Top]