發(fā)布時(shí)間：2024-06-07 04:34

　　在本文中,我們提出一些新的高效的解析方法來(lái)求解金融,應(yīng)用物理科學(xué)和工程中一些重要的分?jǐn)?shù)階(非整數(shù))和非分?jǐn)?shù)階(整數(shù))模型,包括Cantor集上出現(xiàn)的不可微問(wèn)題.本文討論了所提方法的具體推導(dǎo)步驟及其收斂性分析和誤差估計(jì).所有提出的解析方法都可應(yīng)用于金融與工程領(lǐng)域的一些實(shí)際模型,如分?jǐn)?shù)階和非分?jǐn)?shù)階擴(kuò)散方程,分?jǐn)?shù)階和非分?jǐn)?shù)階熱方程,分?jǐn)?shù)階Black-Scholes期權(quán)定價(jià)方程,分?jǐn)?shù)階和非分?jǐn)?shù)階波動(dòng)方程,不可微熱方程,波方程和擴(kuò)散方程.在第二章中,我們重點(diǎn)介紹了論文所需的基礎(chǔ)只是,并簡(jiǎn)要回顧了現(xiàn)存文獻(xiàn)中整數(shù)階和非整數(shù)階導(dǎo)數(shù)和積分的歷史.在分?jǐn)?shù)階導(dǎo)數(shù)中,我們簡(jiǎn)要討論了著名的Caputo和Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)和積分.此外,我們還討論了分?jǐn)?shù)階微積分的一些最新進(jìn)展,如Caputo-Fabrizio和Atangana-Baleanu分?jǐn)?shù)階導(dǎo)數(shù),其中Caputo-Fabrizio和Atangana-Baleanu是具有非奇異核的新型分?jǐn)?shù).在第七章中,我們成功地將Caputo-Fabrizio和Atangana-Baleanu分?jǐn)?shù)階導(dǎo)數(shù)以及Laplace型積分變換應(yīng)用于金融中分?jǐn)?shù)階B...

【文章頁(yè)數(shù)】：265 頁(yè)

【學(xué)位級(jí)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(xué)位論文評(píng)閱及答辯情況表



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