Caputo Hadamard Research Articles

Extreme values of solution of Caputo–Hadamard uncertain fractional differential equation and applications

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

Analysis of controllability in Caputo–Hadamard stochastic fractional differential equations with fractional Brownian motion

Analysis of controllability in Caputo–Hadamard stochastic fractional differential equations with fractional Brownian motion

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.

Solvability of nonlinear pseudo‐parabolic equations involving generalized Caputo fractional derivatives

This study considers nonlinear fractional pseudo parabolic equations, which include the generalized Caputo fractional derivatives of a function with respect to an appropriate function, with general nonlocal initial conditions. Here, the fractional derivative is generalized from many well‐known ones, such as the Caputo, Caputo–Katugampola, Caputo–Hadamard, Erdélyi–Kober, and Liouville–Caputo derivatives. We propose sufficient conditions to ensure that the problem has at least one or a unique mild solution. Furthermore, we investigate the continuous dependence of the mild solutions on the fractional order and other inputs. Particularly, source functions in this study may have temporal singularities. Finally, we provide numerical experiments to illustrate and confirm our theoretical findings.

Two regularization methods for identifying the source term of Caputo–Hadamard time‐fractional diffusion equation

In this paper, the inverse problem for identifying the unknown source on time‐fractional diffusion equation with Caputo–Hadamard derivative is considered. This problem is ill‐posed, and two regularization methods are used to solve it. Firstly, we prove that the problem is ill‐posed and give the conditional stability and the optimal error bound. Secondly, the error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under a priori and a posteriori parameter selection rules are given, respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.

Existence of Monotone Positive Solutions for Caputo–Hadamard Nonlinear Fractional Differential Equation with Infinite-Point Boundary Value Conditions

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.

A method for solving Caputo–Hadamard fractional initial and boundary value problems

In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order , and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models.

Ulam–Hyers stability for an impulsive Caputo–Hadamard fractional neutral stochastic differential equations with infinite delay

This paper addresses existence and Ulam–Hyers stability (UHS) problems for an impulsive Caputo–Hadamard fractional neutral functional stochastic differential equation with infinite delay (FNFSDEwID). We first prove the existence and uniqueness of the solution using Banach fixed point theorem and standard stochastic analysis techniques. We then tackle the UHS under a Lipschitz condition on a bounded and closed interval. We end up with an illustrative example that corroborates our theoretical findings.

Ulam type stability for Caputo–Hadamard fractional functional stochastic differential equations with delay

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples.

Some results for a class of delayed fractional partial differential equations with Caputo–Hadamard derivative

In this paper, we study the existence and uniqueness of the global solution to the Fractional Hyperbolic Systems of Partial Differential equations (FHyPDifS) by the Caputo–Hadamard fractional‐order derivative. In addition, we give a new sufficient conditions for the finite time stability (FTS) of FHyPDifS for the Caputo–Hadamard fractional‐order derivative. Finally, a numerical example is presented to illustrate our theoretical results.

Discretised general fractional derivative

A generalised fractional derivative (the ψ-Caputo derivative) is studied. Generalisations of standard discretisations are constructed for this derivative: L1, L1-2, L2-1σ for derivatives of order α∈(0,1), and L2, H2N2, L21 for derivatives of order α∈(1,2). These new discretisations extend known results for the standard Caputo derivative, the Caputo–Hadamard derivative, etc. Numerical examples are given to demonstrate their performance.

On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation

In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. We expand the classical Khasminskii approach to Caputo–Hadamard fractional stochastic equations by analyzing systems solutions before and after applying averaging principle. We provided an applied example that explains the desired results to us.

Sequential Caputo–Hadamard Fractional Differential Equations with Boundary Conditions in Banach Spaces

We present the existence of solutions for sequential Caputo–Hadamard fractional differential equations (SC-HFDE) with fractional boundary conditions (FBCs). Known fixed-point techniques are used to analyze the existence of the problem. In particular, the contraction mapping principle is used to investigate the uniqueness results. Existence results are obtained via Krasnoselkii’s theorem. An example is used to illustrate the results. In this way, our work generalizes several recent interesting results.

An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

This paper is devoted to investigating a numerical scheme for solving the Caputo–Hadamard uncertain fractional differential equations (UFDEs) arising from nonlinear uncertain dynamic systems. In our approach, we define an α-path, which is a link between a Caputo–Hadamard UFDE and a Caputo–Hadamard fractional differential equation and is the inverse uncertainty distribution of a Caputo–Hadamard UFDE. Then, a formula for calculating the expected value of the Caputo–Hadamard UFDE is studied. With the help of the modified predictor–corrector method, some numerical algorithms for the inverse uncertainty distribution and the expected value of the solution of Caputo–Hadamard UFDEs are designed. Corresponding numerical examples are given to confirm the validity and accuracy of the proposed algorithms.

Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order

In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.

Numerical approximation and error analysis for Caputo–Hadamard fractional stochastic differential equations

Numerical approximation and error analysis for Caputo–Hadamard fractional stochastic differential equations

Qualitative Properties of Positive Solutions of a Kind for Fractional Pantograph Problems using Technique Fixed Point Theory

The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions.

New Van der Pol–Duffing Jerk Fractional Differential Oscillator of Sequential Type

The subject of this paper is the existence, uniqueness and stability of solutions for a new sequential Van der Pol–Duffing (VdPD) jerk fractional differential oscillator with Caputo–Hadamard derivatives. The arguments are based upon the Banach contraction principle, Krasnoselskii fixed-point theorem and Ulam–Hyers stabilities. As applications, one illustrative example is included to show the applicability of our results.

Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations

We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh logtj=loga+logtNajN,j=0,1,2,…,N with a≥1, where loga=logt0<logt1<…<logtN=logT is a partition of [loga,logT]. The error estimates under the different smoothness properties of the solution y and the nonlinear function f are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results.

Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification

A finite element scheme for solving a two-timescale Hadamard time-fractional equation is discussed. We prove the error estimate without assuming the smoothness of the solution. In order to invert the fractional order, a finite-element Levenberg–Marquardt method is designed. Finally, we give corresponding numerical experiments to support the correctness of our analysis.