ON NUMERICAL SOLUTIONS OF SOME FRACTIONAL DIFFERETIAL EQUATIONS WITH DELAY

Abstract

This thesis presents some new numerical techniques that endow with improved order of conver gence (in spatial and temporal dimension) and error to the problem of fourth-order fractional sub-diffusion equations with delay. In particular, two-different numerical schemes are presented to deal with various kinds of generalized fourth-order fractional differential equations. These two numerical schemes utilize L2 − 1σ formula, linear operator, and parametric quintic spline operator for increasing the order of convergence in both spatial and temporal dimensions and decreasing the error. Several numerical experimental results are presented in the form of tables and graphs to support the numerical schemes proposed in the thesis. Apart from it, a detailed analysis has been carried out in concert with the comparisons of some existing approaches to highlight the applicability and the virtue of the proposed numerical schemes. We also analyzed solvability, stability, and convergence of numerical schemes proposed. The considered fourth order fractional differential equations have many applications in different imaging areas, in the model of elastic mediums and stress-strain behaviors to discuss eigenfunctions for torsional model/dielectric spherical shells which further leads to fractional distributed-order differential equations, creep or relaxation in visco-elastoplastic materials, control problems, plasma physics, diffusion process models, etc. The work presented in the thesis comprises of seven Chapters. Chapter 1 presents a gen eral introduction to fractional differential equations and the purpose of developing numerical i ii techniques for fractional differential equations. The motivation of the proposed work is also expressed. A brief inspection of the existing numerical methods related to fractional differen tial equations of fourth order for fractional-order α ∈ (0,1) is summed up as a section of the Chapter. In Chapter 2, a compact difference scheme of second-order temporal convergence for fractional sub-diffusion fourth-order neutral delay differential equations is discussed. In this regard, a difference scheme combining the linear operator for spatial discretization along with the L2−1σ formula (a discrete approximation of Caputo fractional derivative) is implemented. Unique solv ability, stability, and convergence of the proposed scheme are proved using the discrete energy method in L2 norm. Established scheme is of second-order convergence in time and fourth-order convergence in spatial dimension i.e. O(τ3−α +h4), where τ and h are time and space step sizes respectively and α ∈ (0,1). Finally, some numerical results are given to show the efficiency and accuracy of our theoretical results. Chapter 3 presents a linearized compact difference scheme for the fourth-order nonlinear frac tional sub-diffusion equation with time delay and variable coefficients. The primary purpose of the Chapter is to use the idea of the L2−1σ formula for temporal dimension and linear operator for spatial dimension. The proposed method is stable and convergent to the analytical solution with the order of convergence O(τ2 +h4), where τ and h are temporal and spatial step lengths, respectively. Numerical experimentation is carried out to show the efficiency and accuracy of the proposed scheme. In Chapter 4 a linearized second-order numerical technique for nonlinear fourth-order distributed fractional sub-diffusion equation with time delay is presented. Time fractional derivative is rep resented using Caputo derivative and further approximated using L2−1σ formula, which gives the second-order temporal convergence and linear operator is employed for spatial dimensions.The proposed method is stable and convergent to the analytical solution with the order of con vergence O(τ2 + h4 + (∆α)4) and shows the improvement in comparison to existing schemes. Nonlinear terms are linearized with the help of Taylor’s series. At last, we provided a few exam ples to show the efficiency of the compact difference scheme to support the theoretical results and also presented a comparison with L1-approximation of Caputo fractional derivative. In Chapter 5 a numerical scheme for neutral fractional differential equation of fourth-order with variable coefficients is considered. This Chapter is a generalization of Chapter 2. The proposed numerical scheme gives O(τ2+h4) convergence for temporal and spatial dimensions. Numerical stability, solvability and convergence of the constructed scheme are proved. Chapter 6 presents a numerical scheme for fractional variable-order differential equations of fourth-order with delay. Here, we propose to use a parametric quintic spline operator in the spatial dimension and L2−1σ formula for the time dimension. The stability, convergence, and solvability are rigorously proved using a discrete energy method. Our proposed scheme improves convergence in both aspects (spatial-dimension and time-dimension) in comparison to those ear lier work. Numerical simulation is carried out using the MATLAB software to demonstrate the effectiveness of our scheme. Chapter 7 has the concluding remarks of the overall work proposed in the thesis. In particular, the advantages of the proposed algorithms are given. Apart from it, several directions for the future scope of the work presented in the thesis is also provided.