Approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.

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“…In this subsection, we give a brief introduction of fractional calculus theory and some initial results of them that are used for this paper …”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…In this subsection, we give a brief introduction of fractional calculus theory and some initial results of them that are used for this paper. [28][29][30][31] Definition 1. A real function f(x), x > 0 is said to be in the space C , ∈ R, if there exists a real number p > such that…”

Section: Fractional Calculusmentioning

confidence: 99%

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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“…where U = [u 0 , u 1 , …, u n ] T , and u i = u(ih), i = 0, 1, …, n. By considering Equations (4) and (5), and by expanding entries of P(x)P T (x) by 1D-PBPFs, we obtain…”

Section: D-pbpfs and Their Propertiesmentioning

confidence: 99%

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

Mirzaee

Alipour

2018

Numerical Methods Partial

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In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.

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“…Quadratic integral equations have a wide application in several problems of chemical and mathematical physics, including the kinetic theory of gasses, the theory of radiative transfer, and the theory of traffic. Because of the widespread use of these types of equations and the impossibility of solving many of them analytically, their approximate solution by numerical methods has been considered by many researchers which can be pointed to Picard and Adomian methods, repeated trapezoidal methods, and operational matrix method …”

Section: Introductionmentioning

confidence: 99%

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient. KEYWORDS collocation method, cubic and bicubic B-spline functions, nonlinear quadratic integral equations, stochastic integral equations MSC CLASSIFICATION 60H20; 97N50; 65D30; 41A15; 65D05 wileyonlinelibrary.com/journal/mma

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Approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions

Cited by 29 publications

References 17 publications

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

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