在本文中,我們提出一些新的高效的解析方法來求解金融,應用物理科學和工程中一些重要的分數階(非整數)和非分數階(整數)模型,包括Cantor集上出現(xiàn)的不可微問題.本文討論了所提方法的具體推導步驟及其收斂性分析和誤差估計.所有提出的解析方法都可應用于金融與工程領域的一些實際模型,如分數階和非分數階擴散方程,分數階和非分數階熱方程,分數階Black-Scholes期權定價方程,分數階和非分數階波動方程,不可微熱方程,波方程和擴散方程.在第二章中,我們重點介紹了論文所需的基礎只是,并簡要回顧了現(xiàn)存文獻中整數階和非整數階導數和積分的歷史.在分數階導數中,我們簡要討論了著名的Caputo和Riemann-Liouville分數階導數和積分.此外,我們還討論了分數階微積分的一些最新進展,如Caputo-Fabrizio和Atangana-Baleanu分數階導數,其中Caputo-Fabrizio和Atangana-Baleanu是具有非奇異核的新型分數.在第七章中,我們成功地將Caputo-Fabrizio和Atangana-Baleanu分數階導數以及Laplace型積分變換應用于金融中分數階B...

【文章頁數】：265 頁

【學位級別】：博士

【文章目錄】：
摘要
Abstract
附件
Chapter 1 Introduction
Chapter 2 Brief history of fractional calculus
    §2.1 Integer and Non-integer Order Derivatives
        §2.1.1 Basic Definitions of Non-integer Order Derivative
        §2.1.2 Some Basic Definition of Fractional Integrals
Chapter 3 Integral transform and their applications
    §3.1 New Integral Transforms for Solving Ordinary and Partial DifferentialEquations
    §3.2 J-transform Properties and Its Applications
        §3.2.1 Applications of J-transform to Partial Differential Equations
        §3.2.2 Applications of J-transform to Ordinary Differential Equations
        §3.2.3 Is J-transform more efficient than the Sumudu transform and the natural transform?
        §3.2.4 Is J-transform more efficient than the Laplace transform?
    §3.3 Shehu Transform Properties and Its Applications
        §3.3.1 Properties of Shehu Integral transform
        §3.3.2 Applications of Shehu transform to Ordinary Differential Equations
        §3.3.3 Applications of Shehu transform to Partial Differential Equations
    §3.4 Background of Fuzzy Function and Fuzzy sets
    §3.5 Fuzzy Shehu Transform and Its Applications
        §3.5.1 Some Basic Properties of Fuzzy Shehu Integral Transform
        §3.5.2 Application of Fuzzy Shehu Transform to Second-Order Fuzzy Initial Value Problem
        §3.5.3 Application of Fuzzy Shehu Transform to Fuzzy Volterra Integral Equation of the Second Kind of the Form
Chapter 4 Fractal models on Cantor sets
    §4.1 Preliminaries of Local Fractional Calculus
        §4.1.1 Local Fractal Derivative
        §4.1.2 Local Fractal Integral
        §4.1.3 Some Integral Transform on Fractal Space
        §4.1.4 Local Fractal Natural Transform and Its Properties
    §4.2 Applications of Local Fractal Natural Transform
        §4.2.1 Application on signal defined on a Cantor sets
        §4.2.2 Application of Non-differentiable Ordinary Differential Equations
        §4.2.3 Application of Non-differentiable Volterra Integral Equation of the Second Kind
        §4.2.4 Application of Non-differentiable Heat Equation Defined on Cantor Sets
        §4.2.5 Application of Non-differentiable Wave Equation Defined on Cantor Sets
Chapter 5 Analytical methods for fractal models
    §5.1 The Homotopy Analysis Method (HAM)
    §5.2 Local Fractional Homotopy Analysis Method
        §5.2.1 Convergence Analysis of the Local Fractional Homotopy Analysis Method
        §5.2.2 Application of the Local Fractional Homotopy Analysis Method to Non-differentiable Fractional Heat Equation
    §5.3 Fractal Laplace Homotopy Analysis Method
        §5.3.1 Convergence Analysis of the Local Fractional Laplace Homotopy Analysis Method
        §5.3.2 Application of the LFLHAM and Its Comparison with LFHAM on Non-differentiable Linear and Nonlinear Fractional Wave E-quations
    §5.4 Fractal Natural Decomposition Method
        §5.4.1 Convergence Analysis of the Local Fractional Natural Decomposition Method
    §5.5 Applications of the LFNDM
Chapter 6 Analytical techniques for fractional models
    §6.1 Homotopy Analysis Shehu Transform Method
        §6.1.1 Convergence Analysis of the Homotopy Analysis Shehu Transform Method
        §6.1.2 Absolute Error Analysis of the HASTM
        §6.1.3 Homotopy Perturbation Laplace Transform Technique (HPLTT)
        §6.1.4 Applications of the Homotopy Analysis Shehu Transform Method to Linear and Nonlinear Fractional Diffusion Equations
    §6.2 Homotopy Analysis Transform Algorithm
        §6.2.1 Convergence Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
        §6.2.2 Error Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
        §6.2.3 Applications of the Homotopy Analysis Fuzzy Shehu Transform Algorithm to Fuzzy Fractional Partial Differential Equations
    §6.3 Homotopy Perturbation Transform Method
        §6.3.1 Application of the Homotopy Perturbation Method Shehu Transform Method to Fractional Models
Chapter 7 Analytical methods for Black-Scholes equation
    §7.1 Homotopy Perturbation Method (HPM)
    §7.2 Analytical solutions for Option Pricing Equation
    §7.3 Application of NHPM on option pricing equation
    §7.4 New fractional option pricing equations
        §7.4.1 Modelling of Fractional Black-Scholes European option pricing equations with Atangana-Baleanu fractional derivative
    §7.5 The Existence and Uniqueness Analysis
    §7.6 New Q-Homotopy Analysis Transform Method
        §7.6.1 Q-Homotopy Analysis Transform Method via Caputo,Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives for Option Pricing Equation in Finance
    §7.7 Application of the q-homotopy analysis method to new fractional option pricing equations
    7.8 Numerical Results and Discussion
Chapter 8 Conslusions
    §8.1 Conslusions
Major Achievement in this Dissertation
Appendix
References
Publications
Acknowledgement
學位論文評閱及答辯情況表



本文編號：3990787