A Study of Efficient Numerical Methods and their Applications in Fractional Differential Equations

Fractional differential equations can only be solved in series because of the difficulty of calculating with fractional operators. A approach that is effective for solving equations with fractional derivatives is the fractional power series method. M. Kurulay and M. Bayram provide a technique for solving fractional PDEs using a power series. The power series form (PSM) is given as follows:

Consider the following compound fractional relaxation equation as follows: with u(0) =1 Let we take solution of (1) in the subsequent series form By comparing coefficients of , we get recurrence relation to get all information about constants. If we put in eq. (1), we get On solving, we get The above result is 8th approximation results of (1) when Advantage of power series method to freedom of choosing any point in the interval of integration with approximate solution.

Using Laplace transforms, Guo-Cheng Wu and Dumitru Baleanu developed a unique variational iteration approach that incorporates Lagrange multipliers. Nonlinear fractional derivative problems occur often in mathematical physics and other related fields, and Lagrange multiplier methods have been extensively employed to solve these equations. He developed the variational iteration method for solving wide range of applications. Setting the correlation functional; identifying the Lagrange multipliers; setting an initial iteration are all phases in this approach. Poor convergence occurs when the approach is applied to ordinary differential equations using the Lagrange multiplier. For fractional differential equations with starting values, the modified VIM technique has solved this issue by defining a Lagrange multiplier from the Laplace transform. To further understand the MVIM approach, let's look at the following nonlinear differential problem. with the initial conditions for k = 0,1, 2,.....,m =1,where is a linear and N1 is a nonlinear bounded operator, is known as regular function . Now take Laplace transform of equation (5), to construct correction functional is From (8), we calculate Lagrange multiplier as The successive approximations can be given by taking inverse Laplace transform of equation

(7)

with initial conditions

We now apply MVIM to solve following composite fractional equation subjected to the conditions According to MVIM method the correction function for equation (11) has constructed as follows: Case I: When n = 0, above equation gives first iteration. Case II: For n = 1, yields

In this section, MVIM method is applied to solve following equation which is third approximate solution. For α = 1, in above equation yields Since (13) is an ordinary case problem, we may infer that the third approximation of MVIM's series solution is an exact match with the actual answer, making it obvious that MVIM's accuracy is improved

VIM approach is now used to solve the fractional Susceptible-Infected-Recovered model. Developed by W.O. Kermack, the SIR model has played a significant role in mathematical epidemiology in calculating the population's prevalence of various diseases and the number of persons who have recovered. Fractional model start with some basic notations:

R ( ) indicate quantity of recovered people at time where N is overall population size. Following the assumptions, we use a fractional order SIR model.: In the fractional SIR system (equations (15) to (17)), a sophisticated mathematical technique called the modified VIM approach with opening scenarios was used to achieve an approximate analytical solution .For each fractional ordertime derivative of FSIR, numerical computations are performed, which are illustrated visually. The following is how equation (15)'s correction function was generated using the MVIM method: Case I: when n=0, equations give first iteration Case II: when n=1, we can obtain next components for x( ), y( ), z( ) as Similarly And

In similar manner, rest of components can be obtained and rapidly converges to exact solution as rapidly converges it means only few terms are required to get estimated solutions.

Consider nonlinear fractional derivative equation of the following type Taking order differentiation of (21) where k = 0, 1, 2,.... Let us suppose Putting (23) in (22), we get Putting different coefficients, we get

AEM-2 is useful for solving following form of nonlinear equation where y(0) = 0. Equivalent integral equation of (1) which is in best agreement with exact result merely at second approximate solution. Hence this new method is comparative effective and fast converging method.

In order to show, appraise, and enhance the research objectives outlined above, the most appropriate and powerful arithmetical techniques for nonlinear fractional ordinary differential equations are being developed. Several effective numerical and analytical strategies for fractional derivative equations, both linear and non-linear, are presented and applied in mathematical models. Fractional derivative problems may be solved using the MVIM and power series techniques. In the second iteration, the proposed approach is able to provide results in a series form that converges quickly, and the precise results are acquired. There is a long-term relationship between fractional time derivatives and solution in entire instances, according to the results. The effectiveness of two new efficient approaches is shown by implementing them on a class of nonlinear fractional differential equations. Solving the composite fractional relaxation equation using the fractional power series approach is the focus of this section. In order to solve classical order 1 is illustrated in graphical form. To solve nonlinear fractional equations and fractional order gas dynamics equations, several innovative efficient numerical techniques AEM-1 and AEM-2 are examined.

1. Deshna Loonker and P. K. Banerji, ―Wavelet transform of Fractional integrals for integrable boehmians,‖ Application and applied mathematics (AAM), Vol. 5, No. 1, pp. 1-10, 2012. 2. Virginia Kirykova, ―Multiple (multi index) Mittag-Leffler functions and relations to generalized Fractional Calculus,‖ Journal of computational and applied mathematics, Vol.18, pp. 241-259, 2000. 3. Guido Maione, Member, ―On the Laguerre rational approximation to Fractional discrete derivative and integral operators,‖ IEEE transactions on automatic control, Vol. 58, No. 6, 2013. 4. Nadia Benkhettou, Artur M. C. Brito da Cruz and Delfim F. M. torres, ―A Fractional Calculus on arbitrary time scales: Fractional differentiation and Fractional integration,‖ math.CA, arXiv:1405.2813, Vol. 1, 2014. 5. Sachin Bhalekar and Varsha Daftardar Gejji, ―A predictor-corrector scheme for solving nonlinear delay differential equations of Fractional order,‖ Journal of Fractional Calculus and applications, Vol. 1, No. 5, pp 1-9, 2011. 6. Yanqin Liu, ―Study on multi order Fractional Fokker Planck equation by variational iteration method,‖ Journal of Fractional Calculus and applications, Vol. 2, No. 3, pp. 1-8, 2012. 7. Wendi Bao, Yongzhong Song, ―Multiquadric quasi interpolation methods for solving partial differential algebraic equations,‖ Wiley online library, Vol. 30, Issue 1, pp. 95-119, 2013. 8. Zhang Shuqin, ―Existence of solution for a boundary value problem of Fractional order, Acta mathematica scientia,‖ Vol. 26 (B), pp.220-228, 2006 9. Mujeeb ur Rehman and Rehmat Ali Khan, ―Positive solutions to toupled system of Fractional differential equations,‖ International journal of nonlinear science, zol.10, No.1, pp. 96-104, 2010. 10. Bashir Ahmed and Ahmed Alsaedi, ―Nonlinear Fractional differential equations with nonlinear Fractional integro-differential boundary conditions,‖ Boundary value problems, a Springer open journal, 2012.

Research Scholar, Sunrise University, Alwar, Rajasthan