# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021021
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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

Received  January 2021 Revised  May 2021 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.
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 & Optimization, doi: 10.3934/naco.2021021
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##### References:
Exact and approximate solutions for ($n=3$) Example 5.1
Exact and approximate solution for ($n=4$) Example 5.2
Exact and approximate solutions for ($n=5$) Example 5.3
Exact and approximate solutions and square error for ($n = 3$) Example 5.1
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0.0$ $0.0$ $0.0$ $0.1$ $0.2518$ $0.2001$ $0.042224 \times 10^{-2}$ $0.2$ $0.4637$ $0.5103$ $0.0217156 \times 10^{-1}$ $0.3$ $0.6355$ $0.6858$ $0.0253009 \times 10^{-1}$ $0.4$ $0.7673$ $0.7898$ $0.050625 \times 10^{-2}$ $0.5$ $0.8591$ $0.8479$ $0.012544 \times 10^{-2}$ $0.6$ $0.9109$ $0.8940$ $0.028561 \times 10^{-2}$ $0.7$ $0.9227$ $0.8790$ $0.0190969 \times 10^{-1}$ $0.8$ $0.8946$ $0.8790$ $0.024336 \times 10^{-1}$ $0.9$ $0.8264$ $0.8259$ $0.025 \times 10^{-5}$ $1$ $0.7182$ $0.7603$ $0.0167246 \times 10^{-1}$
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0.0$ $0.0$ $0.0$ $0.1$ $0.2518$ $0.2001$ $0.042224 \times 10^{-2}$ $0.2$ $0.4637$ $0.5103$ $0.0217156 \times 10^{-1}$ $0.3$ $0.6355$ $0.6858$ $0.0253009 \times 10^{-1}$ $0.4$ $0.7673$ $0.7898$ $0.050625 \times 10^{-2}$ $0.5$ $0.8591$ $0.8479$ $0.012544 \times 10^{-2}$ $0.6$ $0.9109$ $0.8940$ $0.028561 \times 10^{-2}$ $0.7$ $0.9227$ $0.8790$ $0.0190969 \times 10^{-1}$ $0.8$ $0.8946$ $0.8790$ $0.024336 \times 10^{-1}$ $0.9$ $0.8264$ $0.8259$ $0.025 \times 10^{-5}$ $1$ $0.7182$ $0.7603$ $0.0167246 \times 10^{-1}$
Exact and approximate solutions and square errors for ($n=4$) Example 5.2
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0.0$ $0.0$ $0.0$ $0.1$ $0.110$ $0.162$ $0.02703\times 10^{-1}$ $0.2$ $0.240$ $0.280$ $0.01600\times 10^{-1}$ $0.3$ $0.390$ $0.419$ $0.03481\times 10^{-2}$ $0.4$ $0.560$ $0.579$ $0.0361\times 10^{-2}$ $0.5$ $0.750$ $0.762$ $0.0144\times 10^{-2}$ $0.6$ $0.960$ $0.965$ $0.025\times 10^{-3}$ $0.7$ $1.190$ $1.186$ $0.016\times 10^{-3}$ $0.8$ $1.440$ $1.426$ $0.0196\times 10^{-3}$ $0.9$ $1.710$ $1.683$ $0.0728\times 10^{-1}$ $1$ $2.000$ $1.957$ $0.01849\times 10^{-1}$
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0.0$ $0.0$ $0.0$ $0.1$ $0.110$ $0.162$ $0.02703\times 10^{-1}$ $0.2$ $0.240$ $0.280$ $0.01600\times 10^{-1}$ $0.3$ $0.390$ $0.419$ $0.03481\times 10^{-2}$ $0.4$ $0.560$ $0.579$ $0.0361\times 10^{-2}$ $0.5$ $0.750$ $0.762$ $0.0144\times 10^{-2}$ $0.6$ $0.960$ $0.965$ $0.025\times 10^{-3}$ $0.7$ $1.190$ $1.186$ $0.016\times 10^{-3}$ $0.8$ $1.440$ $1.426$ $0.0196\times 10^{-3}$ $0.9$ $1.710$ $1.683$ $0.0728\times 10^{-1}$ $1$ $2.000$ $1.957$ $0.01849\times 10^{-1}$
Exact and approximate solutions and square errors for ($n=5$) Example 5.3
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0$ $0$ $0$ $0.1$ $0.316$ $0.371$ $0.01\times10^{-6}$ $0.2$ $0.0594$ $0.0897$ $0.09193\times10^{-5}$ $0.3$ $0.1643$ $0.164465$ $0.01225\times10^{-5}$ $0.4$ $0.2530$ $0.25336$ $0.01296\times10^{-5}$ $0.5$ $0.3535$ $0.35402$ $0.02704\times10^{-5}$ $0.6$ $0.4646$ $0.046525$ $0.04225\times10^{-5}$ $0.7$ $0.5857$ $0.58621$ $0.02601\times10^{-5}$ $0.8$ $0.7155$ $0.71527$ $0.00525\times10^{-5}$ $0.9$ $0.8538$ $0.55476$ $0.09216\times10^{-5}$ $1$ $1$ $1.000684$ $0.00467856\times10^{-2}$
 $x_{i}$ $exact \ solutions$ $approximation \ solutions$ $errors$ $0.0$ $0$ $0$ $0$ $0.1$ $0.316$ $0.371$ $0.01\times10^{-6}$ $0.2$ $0.0594$ $0.0897$ $0.09193\times10^{-5}$ $0.3$ $0.1643$ $0.164465$ $0.01225\times10^{-5}$ $0.4$ $0.2530$ $0.25336$ $0.01296\times10^{-5}$ $0.5$ $0.3535$ $0.35402$ $0.02704\times10^{-5}$ $0.6$ $0.4646$ $0.046525$ $0.04225\times10^{-5}$ $0.7$ $0.5857$ $0.58621$ $0.02601\times10^{-5}$ $0.8$ $0.7155$ $0.71527$ $0.00525\times10^{-5}$ $0.9$ $0.8538$ $0.55476$ $0.09216\times10^{-5}$ $1$ $1$ $1.000684$ $0.00467856\times10^{-2}$
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