The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations

Fractional calculus became a vital tool in describing many phenomena appeared in physics, chemistry as well as engineering fields. Analytical solution of many applications, where the fractional differential equations appear, cannot be established. Therefore, cubic polynomial spline-function-based method combined with shooting method is considered to find approximate solution for a class of fractional boundary value problems (FBVPs). Convergence analysis of the method is considered. Some illustrative examples are presented.

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Construction and Convergence Analysis of Polynomial Spline Function of Degree Eight for Fractional Differential Equations

Journal of Zankoy Sulaimani - Part A ◽

10.17656/jzs.10831 ◽

2020 ◽

Vol 22 (2) ◽

pp. 319-326

Author(s):

Amina H. Ali ◽

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Faraidun K. Hamasalh ◽

Balen D. Yasin ◽

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...

Keyword(s):

Differential Equations ◽

Convergence Analysis ◽

Fractional Differential Equations ◽

Spline Function ◽

Polynomial Spline ◽

Fractional Differential

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Green–Haar wavelets method for generalized fractional differential equations

Advances in Difference Equations ◽

10.1186/s13662-020-02974-6 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Mujeeb ur Rehman ◽

Dumitru Baleanu ◽

Jehad Alzabut ◽

Muhammad Ismail ◽

Umer Saeed

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Haar Wavelet ◽

Haar Wavelets ◽

Computational Time ◽

Operational Matrices ◽

Numerical Techniques ◽

Fractional Differential ◽

Fractional Boundary Value Problems ◽

Better Than

Abstract The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

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Positive solutions for a system of nonlocal fractional boundary value problems

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-013-0061-4 ◽

2013 ◽

Vol 16 (4) ◽

Cited By ~ 32

Author(s):

Johnny Henderson ◽

Rodica Luca

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Positive Solutions ◽

Fractional Differential Equations ◽

Existence Results ◽

Hammerstein Integral Equations ◽

Existence Of Positive Solutions ◽

Multipoint Boundary Conditions ◽

Fractional Differential ◽

Fractional Boundary Value Problems

AbstractWe investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to multipoint boundary conditions. Existence results for systems of nonlinear Hammerstein integral equations are also presented. Some nontrivial examples are included.

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Numerical Solution for a System of Fractional Differential Equations with Applications in Fluid Dynamics and Chemical Engineering

International Journal of Chemical Reactor Engineering ◽

10.1515/ijcre-2017-0093 ◽

2017 ◽

Vol 15 (5) ◽

Cited By ~ 2

Author(s):

Bijil Prakash ◽

Amit Setia ◽

Shourya Bose

Keyword(s):

Fluid Dynamics ◽

Exact Solution ◽

Approximate Solution ◽

Differential Equations ◽

Numerical Solution ◽

Fractional Differential Equations ◽

Crucial Role ◽

Chemical Engineering ◽

Haar Wavelets ◽

Fractional Differential

Abstract In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

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