Approximate analytical and numerical methods for defferential equations of fractional order

Abstract

Recently, Fractional Calculus (FC) has seen multiple real-world applications in the fields of physics, biology, engineering, and finance. Numerous scholars have devised various numerical and approximate techniques to solve nonlinear fractional differential equations (NFDEs). However, analytic solutions to FDs usually is not available, especially for nonlinear problems. The most common definitions of fractional derivatives are Riemann-Liouville (RL), Liouville-Caputo (LC), Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). The CF introduced a new derivative which does not have a singular kernel uses exponential decay as the kernel instead of the power law. Thereafter, the AB introduced a new differential and integral operator that includes the generalized Mittag-Leffter (ML) function. This operator is highly contributive in modeling real world problems due to its nonsingularity and nonlocality. The main contribution of this thesis is the demonstration of how to apply fractional homotopy analysis transform method (FHATM) for different fractional derivatives. Special attention has been paid to AB fractional derivative due to its important applications. In addition, a new class of predictor-corrector method for solving nonlinear fractional initial value problems (NFIVPs) has been proposed with increase accuracy. This thesis can be considered as three parts, the first part considered the Human Immunodeficiency Virus (HIV) model and we extend this model using fractional order derivatives considering by the AB fractional derivative in Caputo sense (ABC). This model is analysed rigorously and solved numerically using FHATM and Toufik-Atangana Method (TAEM).The graphical results reveal that the model with fractional derivatives give useful and biologically more feasible consequences. The second part considered the Gray-Scott model (GSM) and we extend this model using three different fractional operators namely; LC, CF and AB operators. The FHATM technique is used to derive solutions for GSM. The solutions obtained were compared with the exact solution, which were in excellent agreement. The third part in this thesis proposed fractional high order numerical method for integrating the nonlinear fractional differential equations. We get corrector formula with high accuracy which is implicit as well as predictor formula, which is explicit and has the same precision order as the corrective formula. On the other hand, the so-called memory term is computed only once for both prediction and correction phases, which indicates the low computational cost of the proposed method. Also, the error bound of the proposed numerical scheme is offered. The error analysis for illustrative examples have shown that this method is more efficient and accurate compared with other existing predictor-corrector methods (PCMs).