Analytical and Approximate Methods for Fractional Differential Equations and their Applications

Abstract

Fractional calculus can be treated as generalization of conventional calculus in the newlinesense that it extends the concept of the derivatives and integrals to include arbitrary orders. newlineEffective mathematical modelling by fractional order differential equation requires the newlinedevelopment of reliable and flexible numerical methods. newlineAt first the efficient numerical methods for non-linear fractional ordinary differential newlineequation have been reviewed and implemented viz Modified Variation Iteration Method newline(MVIM), Mittag-Leffler method and Homotopy Perturbation Method respectively on: newlinei. Analytical solution for fractional Glucose Insulin Regulatory Model with the newlinediscussion of Equilibrium States and Stability Analysis. newlineii. The numerical solution of novel Fractional Biochemical Reaction model is also newlineobtained by implementation of the Mittag-Leffler method. newlineiii. The development of fractional dynamical population model and its numerical newlinesolution is determined by using Homotopy Perturbation Method. newlineThen we introduced novel analytical approximate method Homotopy Perturbation and newlineNatural Transform Method (HPNT) for solution of non - linear Fractional differential newlineequations and implemented the novel method HPNT for the solution of Fractional Heat newlineequation, Fractional Fokker Planck equation, Fractional Burger equation, Fractional Klein newlineGorden equation and Fractional KdV equation. Further their solutions behaviour are newlinedepicted by graphs for various fractional order. Then we employed Caputo Fabrizio newlinefractional derivative operator in formulating the classical order influenza model to the newlinefractional order SLIA mathematical model. Numerical technique Iterative Laplace newlinetransform is employed to carry out the analytic approximate solution of the fractional newlineSLIA model.At last we demonstrated the numerical efficiency of the proposed HPNT analytic newlineapproximate technique through error analysis and we carried out the comparison of HPNT newlinewith other numerical methods