American scientific publishing group

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power $q=\frac{1}{m},m\in Z^{+}$, where the fractional derivative of order $\alpha =q\gamma$, for which $\gamma \in Z^{+}$. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

In this paper, we conduct a numerical study of the Sturm–Liouville eigenvalue problem using the $$\operatorname {sinc}$$ sinc -Galerkin method. We provide a comprehensive discussion of the fundamental properties of $$\operatorname {sinc}$$ sinc methodology and its advantages. The solution is expressed as a finite expansion in terms of composite translated $$\operatorname {sinc}$$ sinc functions, whose coefficients are subsequently determined. We started by expressing the original Sturm–Liouville problem in a bilinear form relative to a predefined basis, namely the $$\operatorname {sinc}$$ sinc basis, with corresponding coefficients that can be determined. We then represented this bilinear form using a set of suitable integrals. Finally, conformal mapping and its inverse were systematically evaluated at $$\operatorname {sinc}$$ sinc grid points by employing the $$\operatorname {sinc}$$ sinc quadrature rule. We find that the result in error converges exponentially to zero. Three test cases with applications in the physical sciences were used to demonstrate the methodology. The findings demonstrate that the approach is highly effective, practical, and applicable to a wide range of issues. The study demonstrates that $$\operatorname {sinc}$$ sinc -Galerkin has an accuracy of order $$\mathcal {O}(\exp (-c \sqrt{N}))$$ O ( exp ( - c N ) ) and rapidly converges to the exact solution.

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.

This work presents a general framework for solving generalized fractional differential equations based on operational matrices of the generalized Bernstein polynomials. This method effectively obtains approximate analytical solutions of many fractional differential equations. The generalized fractional derivative of the Caputo type with two parameters and its properties are studied. Using orthonormal Bernstein polynomials has led to the development of fractional polynomials, which offer an approximate solution for ordinary fractional differential equations. The approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their operational matrices for fractional integration and derivative in the generalized Caputo sense to achieve this objective. Operational matrices convert ordinary differential equations into a system of algebraic equations, which can be solved using Newton’s method. The convergence analysis and error estimate associated with the proposed problem have been investigated using the approximation of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters of the fractional derivative is presented in several examples. Finally, several examples are included to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal Bernstein polynomials (FOBPs) approximation.