Types Of Fractional Differential Equations Research Articles

The Averaging Principle for Caputo Type Fractional Stochastic Differential Equations with Lévy Noise

The Averaging Principle for Caputo Type Fractional Stochastic Differential Equations with Lévy Noise

Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

This paper proposes a scientific machine learning approach based on Deep Physics Informed Neural Network (PINN) to solve ψ-Caputo-type differential equations. The trial solution is constructed based on the Theory of Functional Connection (TFC), and the loss function is built using the L1-based difference and quadrature rule. The learning is handled using the new hybrid average subtraction, standard deviation-based optimizer, and the nonlinear least squares approach. The training error is theoretically obtained, and the generalization error is derived in terms of training error. Numerical experiments are performed to validate the proposed approach. We also validate our scheme on the SIR model.

A novel fixed point iteration process applied in solving the Caputo type fractional differential equations in Banach spaces

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized [Formula: see text]-pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature.

Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

Using the Elzaki decomposition method to solve nonlinear fractional differential equations with the Caputo-Fabrizio fractional operator

يتم تطبيق تقنيات حساب التفاضل والتكامل الجزئي بنجاح في العديد من فروع العلوم والهندسة ، احدى هذه التقنيات طريقة تحليل الزكي وادوميان (EADM) ، والتي لم يدرسها الباحثون باستخدام المشتق الكسري لـ Caputo-Fabrizio. يهدف هذا العمل لدراسة طريقة تحليل الزكي وادوميان (EADM) لحل المعادلات التفاضلية الكسرية بمشتق Caputo-Fabrizio. قدمنا خوارزمية هذه الطريقة مع المؤثر CF وناقشنا تقاربها باستخدام طريقة سلسلة كوشي بعد ذلك ، تم تطبيق الطريقة لحل معادلات برجر، الشبية بالحرارة ، و برجر المزدوجة مع المؤثر Caputo -Fabrizio. في الختام أن الطريقة كانت متقاربة وفعالة لحل هذا النوع من المعادلات التفاضلية الكسرية.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

Nonlinear implicit generalized Hilfer type fractional differential equations with non-instantaneous impulses

Nonlinear implicit generalized Hilfer type fractional differential equations with non-instantaneous impulses

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J,S)-fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

Large Fractional Linear Type Differential Equations

This paper aims to handle some types of fractional differential equations with a fractional-order values β>1. In particular, we propose a novel analytical solution called an atomic solution for certain fractional linear type differential equations as well as for some other types of partial differential equations with fractional-order values exceeding one. Some examples are provided to validate our findings.

The Study of Bicomplex-Valued Controlled Metric Spaces with Applications to Fractional Differential Equations

In this paper, we introduce the concept of bicomplex-valued controlled metric spaces and prove fixed point theorems. Our results mainly focus on generalizing and expanding some recently established results. Finally, we explain an application of our main result to a certain type of fractional differential equation.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods.

Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials

The theory of mixed fractional operators is still an uncovered area in fractional modelling. These multi-sided operators result by combining two fractional derivatives with different kernels, that is, the right-sided Caputo's and the left-sided Riemann–Liouville's fractional operators in a sequential manner. Although it may capture different memory phenomenons, there is no general approach to approximate the solution of the mixed type fractional differential equations (MFDEs). In this article, the authors introduce a novel numerical technique that relies on obtaining the matrix representation of the mixed operators. This was done by considering a new extension of the shifted Chebyshev polynomials of the fifth-kind (SFCPs) with a variable domain as a basis of space. Then we employ the spectral collocation method together with the operational matrix to transform the MFDE into the corresponding system of algebraic equations. The convergence analysis of the proposed technique was studied in terms of the generalized Taylor's formula and the Gram determinant. Our findings facilitate further development of this theory by providing such a method of approximation.

Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects

Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order β∈(0,1) are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In Hδ,r, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in Jr. Moreover, the regularity and existence of classical solutions to the equations in Jr. are established and presented.

Analysis of p‐Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term

This paper investigates a new class of p‐Laplacian boundary value problems of Hadamard type fractional differential equations. We emphasize that few works on Hadamard fractional boundary value problems include p‐Laplacian Hadamard type fractional differential equations with nonlinear terms involving Hadamard fractional derivatives of unknown functions. By means of the five functional fixed point theorem, multiple positive solutions are established for the p‐Laplacian Hadamard fractional boundary value problem. An example is constructed for illustration of the main result.

STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study a coupled system of Caputo type fractional differential equations with integral boundary conditions. By Leray-Schauder alternative theorem, the existence of solutions for the factional differential system are obtained. The Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. The main results are well illustrated with examples and numerical simulation graphs. The interesting point of this article is that it not only gives approximate graphs of solution by using the iterative methods, but also verifies the Hyers-Ulam stability of the coupling system by numerical simulation.

Existence of positive solutions for a p-Laplacian type fractional differential equation with boundary conditions

Existence of positive solutions for a p-Laplacian type fractional differential equation with boundary conditions

Applicability of Mönch’s Fixed Point Theorem on a System of (k, ψ)-Hilfer Type Fractional Differential Equations

In this article, we study a system of Hilfer (k,ψ)-fractional differential equations, subject to nonlocal boundary conditions involving Hilfer (k,ψ)-derivatives and (k,ψ)-integrals. The results for the mentioned system are established by using Mönch’s fixed point theorem, then the Ulam–Hyers technique is used to verify the stability of the solution for the proposed system. In general, symmetry and fractional differential equations are related to each other. When a generalized Hilfer fractional derivative is modified, asymmetric results are obtained. This study concludes with an applied example illustrating the existence results obtained by Mönch’s theorem.

Stochastic averaging for a type of fractional differential equations with multiplicative fractional Brownian motion.

This paper focuses on the averaging principle of Caputo fractional stochastic differential equations (SDEs) with multiplicative fractional Brownian motion (fBm), where Hurst parameter 1/2<H<1 and the integral of fBm as a generalized Riemann-Stieltjes integral. Under suitable assumptions, the averaging principle on Hölder continuous space is established by giving the estimate of Hölder norm. Specifically, we show that the solution of the original fractional SDEs converges to the solution of the proposed averaged fractional SDEs in the mean square sense and gives an example to illustrate our result.