A Method to Solve One-dimensional Nonlinear Fractional Differential Equation Using B-Polynomials
Md Habibur Rahman, Muhammad I Bhatti* and Nicholas Dimakis
University of Texas Rio Grande Valley, Edinburg, Texas, USA
*Corresponding Author: Muhammad I Bhatti, University of Texas Rio Grande Valley, Edinburg, Texas, USA.
September 27, 2022; Published: October 31, 2022
>In this article, the fractional Bhatti-Polynomial bases are applied to solve one-dimensional nonlinear fractional differential equations (NFDEs). We derive a semi-analytical solution from a matrix equation using an operational matrix which is constructed from the terms of the NFDE using Caputo's fractional derivative of fractional B-polynomials (B-polys). The results obtained using the prescribed method agree well with the analytical and numerical solutions presented by other authors. The legitimacy of this method is demonstrated by using it to calculate the approximate solutions to four NFDEs. The estimated solutions to the differential equations have also been compared with other known numerical and exact solutions. It is also noted that for solving the NFDEs, the present method provides a higher order of precision compared to the various finite difference methods. The current technique could be effortlessly extended to solving complex linear, nonlinear, partial, and fractional differential equations in multivariable problems.
Keywords: Fractional B-Polynomials; Fractional Differential Equations; Nonlinear Partial Fractional Differential Equation; FRACTIONAL B-polynomials in Multiple Variables
- S G Samko., et al. “Fractional integrals and derivatives: Theory and Applications”. Gordon and Breach Science Publishers (1993).
- A Bekir., et al. “Functional variable method for the nonlinear fractional differential equations”. in AIP Conference Proceedings, (2015): 1648.
- R Hilfer. "Applications of Fractional Calculus in Physics”. Applications of Fractional Calculus in Physics (2000).
- MA Abdou. “An analytical method for space-time fractional nonlinear differential equations arising in plasma physics”. Journal of Ocean Engineering and Science4 (2017).
- Z Dahmani., et al. “The foam drainage equation with time- and space-fractional derivatives solved by the adomian method”. Electronic Journal of Qualitative Theory of Differential Equations (2008): 1-10.
- Podlubny Igor. “Fractional Differential Equations, Volume 198: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution (Mathematics in Science and Engineering)”. Mathematics in Science and Engineering (1999): 198 198.
- M Duarte Ortigueira. “Fractional calculus for scientists and engineers” (2011): 152.
- A Kilbas., et al. “Theory and applications of fractional differential equations” (2006).
- R Hilfer. “Applications of fractional calculus in physics” (2000).
- M Lu and L Wei. “The International Order of Asia in the 1930s and 1950s. Edited by Shigeru Akita and Nicholas J. White. pp. xvii, 308. Aldershot, Ashgate, 2010”. Journal of the Royal Asiatic Society3-4 (2012): 623-624.
- H Naher and F A Abdullah. “New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics”. Journal of the Egyptian Mathematical Society3 (2014): 390-395.
- M A Abdou. “New analytic solution of von karman swirling viscous flow”. Acta Applicandae Mathematicae1 (2010).
- R Cimpoiasu and R Constantinescu. “The inverse symmetry problem for a 2D generalized second order evolutionary equation”. Nonlinear Analysis: Theory, Methods and Applications1 (2010): 147-154.
- R Cimpoiasu and R Constantinescu. “Lie symmetries and invariants for a 2D nonlinear heat equation”. Nonlinear Analysis: Theory, Methods and Applications8 (2008): 2261-2268.
- G Jumarie. “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions”. Applied Mathematics Letters3 (2009): 378-385.
- G Jumarie. “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results”. Computers and Mathematics with Applications9-10 (2006): 1367-1376.
- H Jafari., et al. “A new approach for solving a system of fractional partial differential equations”. Computers and Mathematics with Applications5 (2013).
- A M Spasic and M P Lazarevic. “Electroviscoelasticity of liquid/liquid interfaces: Fractional-order model”. Journal of Colloid and Interface Science1 (2005): 223-230.
- D Baleanu., et al. “Fractional Calculus”. 3 (2012).
- KS Miller and B Ross. “An introduction to the fractional calculus and fractional differential equations”. (1993).
- "The initial value problem for some fractional differential equations with the Caputo derivatives”. Freie Universität Berlin, Fachbereich Mathematik und Informatik, Preprint No. A-98-08.
- YF Luchko and H M Srivastava. “The exact solution of certain differential equations of fractional order by using operational calculus”. Computers and Mathematics with Applications8 (1995): 73-85.
- S Kemple and H Beyer. “Global and causal solutions of fractional differential equations M". SCTP (1997) 210-216.
- D Chouhan., et al. “Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order”. Results in Applied Mathematics 10 (2021).
- K Shah., et al. “Stable numerical results to a class of time-space fractional partial differential equations via spectral method”. Journal of Advanced Research 25 (2000): 39-48.
- M I Bhatti and D D Bhatta. “Numerical solutions of Burgers' equation in a B-polynomial basis”. Physica Scripta (Online)6 (2006): 539-544.
- M I Bhatti. “Solutions of the harmonic oscillator equation in a B-polynomial basis”. Advanced Studies in Theoretical Physics 3 (2009): 9-12.
- M Idrees Bhatti and P Bracken. “Solutions of differential equations in a Bernstein polynomial basis”. Journal of Computational and Applied Mathematics1 (2007): 272-280.
- M I Bhatti., et al. “Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases”. Fractal and Fractional 5 (2016): 106.
- M I Bhatti. “Analytic matrix elements of the schr̈odinger equation”. Advanced Studies in Theoretical Physics1-4 (2013): 11-23.
- M I Bhatti and M H Rahman. “Technique to solve linear fractional differential equations using b-polynomials bases”. Fractal and Fractional4 (2021).
- M Bhatti., et al. “Solution of mathematical model for gas solubility using fractional-order Bhatti polynomials”. Journal of Physics Communications8 (2018): 085013.
- DB Suits. “Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019)”. Journal of the American Statistical Association (1957): 52.280.
- "Wolfram Mathematica: Modern Technical Computing" (2022).
- U Saeed and M U Rehman. “Haar wavelet-quasilinearization technique for fractional nonlinear differential equations”. Applied Mathematics and Computation 220 (2013): 630-648.
- Y LI. “Solving a nonlinear fractional differential equation using Chebyshev wavelets”. Communications in Nonlinear Science and Numerical Simulation 9 (2010).
- Z Odibat and S Momani. “Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order”. Chaos, Solitons and Fractals1 (2008): 167-174.
- J Ha., et al. “26 Results of hyperbolic partial differential equations in B-poly basis”. Journal of Physics Communications9 (2020): 095010.
- A Barari., et al. “Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations”. Acta Applicandae Mathematicae2 (2008): 161-171.
- C Zienkiewicz. “Computational galerkin methods, C. A. J. Fletcher, Springer Verlag, N. Y./Berlin/Heidelberg, 1984 (309 pages $32.90 or DM. 88)”. International Journal for Numerical Methods in Engineering2 (1985): 385-385.