Efficient Computational Approaches and Their Convergence Analysis for Integro-differential Equations with Small Parameters and Fractional Derivatives

Abstract

Integro-differential equations (IDEs) combining both differential and integral terms emerge in diverse fields of science and engineering, for conceptualizing systems that depend on past values of the function. For instance in physics, IDEs depict the conduct of materials possessing memory, like viscoelastic substances. In the field of ecology, these equations find utility in depicting population dynamics, wherein the integral component captures the impact of previous populations on the present state. In the domain of finance, they find application in modeling option valuation and risk management, accounting for the historical trajectory of underlying assets. So, IDEs can be used to model economic systems where past decisions or events can influence the current behavior and this feature of IDEs attracted many researchers to attain insights into a broad array of complex phenomena spanning diverse scientific domains. It is really intricate to find the analytical solutions of the IDEs. So, several ideas are proposed in literature basically consisting of the numerical as well as semi-analytical approaches which are used for finding the solutions to the IDEs. Some of the techniques involves the Adomian decomposition method, homotopy perturbation method, variational iteration method, reproducing kernel method, which are meshfree in nature. Also, there are many numerical techniques such as the finite difference method, finite element method, spectral/collocation method, that are employed to approximate the solution of IDEs. As already mentioned, finding the solutions to IDEs are complicated, but it is even more difficult to obtain the solutions of singular IDEs, where the solution has singularities at some point of the domain. These type of situations often cannot be dealt with the conventional numerical methods, so specially tailored methods are used to deal with such model problems. Researchers have gained insights in solving the regular IDEs but very less contributions are made for analyzing the IDEs with singularities. This thesis intends to develop the higher order numerical methods and semi-analytical methods for finding the solutions of IDEs involving small parameters and fractional derivatives. Also, the convergence analysis/error estimates of all the proposed schemes for various kinds of model problems are broadly discussed. Some numerical simulations are carried out so as to give a pictorial description of how the solution behaves and thereby drawing a validation to the theoretical prospects. Since, we have dealt with two types of IDEs, this thesis is partitioned in two categories, wherein the first five chapters are devoted to find the numerical solutions of singularly perturbed IDEs where the solution undergoes vital changes as the parameter approaches zero. The next three chapters are dedicated to find the approximate/numerical solutions of fractional order IDEs. This dissertation is comprised of a total of ten chapters, of which Chapter 1 deals with the basic ideas of the singularly perturbed differential equations and fractional calculus, inclusive of the important definitions, properties, and the motive of working out various model equations of IDEs. Chapter 2 describes the numerical schemes and their convergence analysis for studying a singularly perturbed Volterra integro differential equation (SPVIDE) using the conventional upwind scheme with the trapezoidal rule for the integral term. The accuracy is further improved by using a post-processing scheme which is validated with a few examples and comparison results. The idea used for the first order SPVIDE is further extended to a second order SPVIDE wherein the kernel is taken to be of the nonlinear type and a hybrid scheme is used for obtaining direct second order convergence. The numerical approach used in Chapter 1 is extended to solve IVP and BVP cases of the singularly perturbed Fredholm integro differential equation (SPFIDE) and singularly perturbed Volterra-Fredholm integro differential equation (SPV-FIDE) in Chapter 3 and Chapter 4 respectively. The system of SPVIDEs is considered in Chapter 5, where the problem is solved using a first order convergent scheme. The numerical approximation is done using the backward Euler method for the differential operator and rectangular rule for the integral operator and with a proper error analysis of the concerned scheme. Further, the scheme is subjected to post-processing technique, thereby giving better accuracy. In Chapter 6, the numerical solution of the singularly perturbed partial differential equation with a Volterra integral is proposed. Firstly, the problem is solved by forming a difference scheme that consists of the implicit-Euler method for the time derivative, an upwind scheme for the spatial derivative, and a left rectangular rule for the integral part that gives a first ordere-uniform convergence. The order of accuracy is then enhanced by successfully applying the Richardson extrapolation technique. Secondly, a direct second order convergent difference scheme is employed, comprising of the Crank-Nicolson scheme in the temporal direction, a hybrid scheme in the spatial direction, and a composite trapezoidal rule in the integral part. Finally, the performance of the numerical schemes is tested by a few examples. The next three chapters of the thesis study the model problems related to fractional IDEs. Chapter 7 focuses on different semi-analytical methods to solve a time fractional partial IDEs (PIDEs). The Adomian decomposition method, the homotopy perturbation method, and the modified homotopy perturbation method are applied to solve the model equation. A brief convergence analysis is carried out for all the proposed techniques and numerical experiments are done to show the efficiency of the suggested methods. In the next chapter, Chapter 8, investigates the fractional order Fredholm-Volterra IDE using both semi-analytical and numerical methods. The semi-analytical approximations are done using the Chebyshev and Bernstein polynomials in the ADM and the numerical scheme is developed using the L1 scheme for the fractional order derivative in combination with appropriate quadrature rules for the integral parts. Some comparisons with the existing results show that the proposed methods are highly productive and reliable. Chapter 9 aims to develop an efficient scheme for a fractional IDE involving a weakly singular kernel. The weakly singular Volterra integral is approached with the production integration rule, and the Caputo order derivative is approximated using the L1 scheme. First order accuracy of this method has been demonstrated. Additionally, the Richardson extrapolation strategy is effectively used to boost the accuracy. Finally, the model problem is also investigated using another scheme wherein, the integral part is discretized using the product trapezoidal rule and fractional derivative is discretized using the L1 scheme which gives better convergence rate in comparison to scheme I. Numerical tests are done to demonstrate the efficacy of the proposed methodologies. Finally, Chapter 10 presents the concluding statements derived from the diverse outcomes discussed in this thesis, along with a handful of potential future avenues for further exploration based on the current research.