Journal ArticleDOI
Solution of Fractional Harmonic Oscillator in a Fractional B-poly Basis
Muhammad I. Bhatti
- Vol. 2, Iss: 2, pp 8
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TLDR
In this paper, an algorithm for approximating solutions to fractional-order differential equations in fractional polynomial basis is presented, where a finite generalized fractional order basis set is obtained from the modified Bernstein polynomials.Abstract:
An algorithm for approximating solutions to fractional-order differential equations in fractional polynomial basis is presented. A finite generalized fractional-order basis set is obtained from the modified Bernstein Polynomials, where α is the fractional-order of the modified Bernstein type polynomials (B-polys). The algorithm determines the desired solution in terms of continuous finite number of generalized fractional polynomials in a closed interval and makes use of Galerkin method to calculate the unknown expansion coefficients for constructing the approximate solution to the fractional differential equations. The Caputo's definition for a fractional derivative is used to evaluate derivatives of the polynomials. Each term in a differential equation is converted into matrix form and the final matrix problem is inverted to construct a solution of the fractional differential equations. However, the accuracy and the efficiency of the algorithm rely on the size of the set of B-polys. Furthermore, a recursive definition for generating fractional B-polys and the analytic formulism for calculating fractional derivatives are presented. The current algorithm is applied to solve the fractional harmonic oscillator problem and a number of linear and non-linear fractional differential equations. An excellent agreement is obtained between desired and exact solutions. Furthermore, the current algorithm has great potential to be implemented in other disciplines, when there are no exact solutions available to the fractional differential equations.read more
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Journal ArticleDOI
Solution of mathematical model for gas solubility using fractional-order Bhatti polynomials
TL;DR: In this paper, an algorithm based on the generalized Galerkin B-poly basis technique was proposed to find a solution of a fractional differential equation in terms of continuous finite number of generalized fractional-order Bhatti polynomial (B-poly) in a closed interval.
Journal ArticleDOI
26 Results of hyperbolic partial differential equations in B-poly basis
TL;DR: In this paper, a two-variable process to estimate results of hyperbolic partial differentiation (HPD) equations in a B-Polynomial (B-Poly) base is established.
Journal ArticleDOI
Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases
TL;DR: In this paper, a multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys).
Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases
TL;DR: In this paper, a multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear FPDE. The Galerkin method was applied to find the agreement between exact and approximate solutions.
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