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Title: Approximation of Fractional Order Differential and Integral Equations Using Integral Transforms and Meshless methods
Authors: Taufiq, Muhammad
Keywords: Physical Sciences
Laplace transform, Local method, Numerical inversion of Laplace transform, Trapezoidal rule, Oscillatory kernels of convolution type, Volterra integral equations, Contour integration, Black-Scholes model, Fractional anomalous sub-di usion equations and Non-linear PDEs.
Issue Date: 2021
Publisher: University of Engineering & Technology Peshawar
Abstract: Integral or di erential equations of fractional order are used to model many problems of science and engineering. These equations have various applications such as viscoelasticity, thermodynamics, uid ow, nance, stochastic models and stock exchange models. The analytical solutions of such type of equations are not easy to obtain, hence discrete approximation methods have to be employed in order to obtain optimized solutions. The fractional derivative is a non-local operator and the discretization of the time fractional derivative is a challenging problem. Time-stepping process and ill-conditioning of the system matrix are the major drawbacks in approximation with global collocation methods. Laplace transform and Meshless methods are well suited to handle these time dependent problems in complex shape domains. The supremacy of these methods, specially in case of fractional order equations, is its non-sensitive nature in time which in contrast to the technique adopted in nite di erence approximation for fractional order operators. In this research work, an e cient numerical scheme based on Laplace transform and local radial basis functions (LT-LRBFs), is designed for the approximation of both time dependent linear and non-linear problems of di erential equations (DEs), linear problems of integral (IEs) and integro-di erential equations (IDEs) of fractional order. The advantages of this method are the resolution of issues of time instability, nonlinearity and ill-conditioning related to the system matrices. In all of the computations numerical inversion of the Laplace transform is required. The procedure of inversion is carried out using contour integration technique with some parabolic or hyperbolic paths. vi In the rst phase of this work, a hybrid approximate scheme is developed in which Laplace transform (LT) and trapezoidal rule with constant step size are used for solution of Volterra integral equations. As operators with fractional order are easily handled through Laplace transform and local radial basis functions (LRBFs), so they are developed for large scale and high dimensional problems. In the second phase, an approximate scheme is designed to approximate time fractional Black-Scholes model (TFBSM) and fractional order anomalous sub-di usion equations (FASDEs). In the third phase, a numerical scheme is constructed in which Laplace transform, local radial basis functions (LRBF) and trapezoidal rule with constant step size are employed in approximation of fractional order partial integro-di erential equations (FPIDEs). The idea is extended for approximation of fractional order non-linear partial di erential equations (FNLPDEs). As the non-linear terms cannot be handled by Laplace transform, so, it is handled by some linearization process coupled with a suitable iterative scheme. In the fourth phase of this research work, an iterative numerical scheme is constructed for the approximation of time fractional non-linear partial differential equations (FNLPDEs). The e ciency, accuracy and stability of the Laplace transform based method is tested by solving problems related to Burgers' equation and KdV equation. The obtained numerical results are in good agreement with exact solutions and other existing methods in literature.
Gov't Doc #: 26817
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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