Efficient Method for the Solution of Fractional-order Differential Equations with Variable Coefficients

In this paper, we propose the Bernoulli wavelet approximation for the solution of the fractional differential equations with variablecoefficients. In the proposed method, the fractional derivatives are transformed using the operational matrix of fractional orderintegration and by doing that differential equation reduces to a system of algebraic equations. The operational matrix of fractional orderintegration is obtained via block pulse functions. Illustrative examples are presented. The examples demonstrate that the method isaccurate and efficient.

PDF

___

Atabakzadeh M.H., Akrami M.H., & Erjaee G.H. (2013) Chebyshev operational matrix method for solving multi order fractional ordinary differential equations. Applied Mathematical Modelling, 37, 8903-8911.
Arikoglu A.,& Ozkol I. (2009), Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, Solitons & Fractals, 40,521-529.
Cajić, M., Karličić D.,& Lazarević M. (2015) Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle. Theoretical and Applied Mechanics, 42(3), 167-190.
Chen D., Chen Y., & Xue D. (2013) Three fractional-order TV-models for image de-noising. Journal of Computer Information Systems, 9 (12), 4773-4780.
Gupta, S., Kumar, D.,& Singh, J.(2015) Numerical study for systems of fractional differential equations via Laplace transform. Journal of the Egyptian Mathematical Society, 23, 256–262.
Hashim I., Abdulaziz O., & Momani S. (2009) Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14, 674–684.
Karaman M.M., Sui Y., Wang H., R.L. Magin, Li Y.,& Zhou X.J.(2016) Differentiating low- and high-grade pediatric brain tumors using a continuous-time random-walk diffusion model at high b-values. Magnetic Resonance in Medicine, 76, 1149-1157.
Karimi H., Moshiri B., Lohmann B., Maralani P. (2005) Haar wavelet-based approach for optimal control of second-order linear systems in time domain, Journal of Dynamical and Control Systems, 11, 237–252.
Khader M.M. (2017), Application of homotopy perturbation method for solving nonlinear fractional heat-like equations using Sumudu transform, Scientia Iranica B, 24, 648-655.
Kilicman A. (2007) Kronecker operational matrices for fractional calculus and some applications. Applied Mathematics and Computation, 187, 250-265.
Lei D., Liang Y., & Xiao R. (2018) A fractional model with parallel fractional maxwell elements for amorphous thermoplastics. Physics A, 490, 465-475.
Li Y., &Yu S.L. (2006)Fractional order difference filters and edge detection. Opto-Electronic Engineering, 33(19), 71-74.
Li Y.L.,& Sun N.(2011) Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Computers & Mathematics with Applications 62 (3),1046-1054.
Mandelbrot B. (1967) Some noises with 1/f spectrum, a bridge between direct current and white noise. IEEE Transactions on Information Theory, 13(2), 289–98.
Moradi G.,& Mehdinejadiani B .(2018) Modeling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection–dispersion equation. Soil and Water Research, 13, 18–28.
Nigmatullin R. R.,. Osokin S.I., & Toboev V.A. (2011) NAFASS: Discrete spectroscopy of random signals. Chaos Solitons Fract, 44, 226-240.
Rahimkhani P., Ordokhani Y. & Babolian E. (2016) An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dynamics, 86, 1649–1661.
Sun H.G., Zhang Y., Chen W. & Reeves D.M. (2014) Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. Journal of Contaminant Hydrology, 157, 47–58.
Tarasova V.V.,& Tarasov V.E. (2017) Logistic map with memory from economic model. Chaos, Solitons and Fractals, 95, 84-91.
Wang X., Qi H., Yu B., Xiong Z., & Xu H. (2017) Analytical and numerical study of electroosmotic slip flows of fractional second grade fluids. Communications in Nonlinear Science and Numerical Simulation, 50, 77-87.