Types Of Fractional Differential Equations Research Articles

APPLICATIONS OF BI-FRAMELET SYSTEMS FOR SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

A New Dynamic Scheme via Fractional Operators on Time Scale

The present work investigates the applicability and effectiveness of the generalized Riemann-Liouville fractional integral operator integral method to obtain new Minkowski, Gr\"{u}ss type and several other associated dynamic variants on an arbitrary time scale, which are communicated as a combination of delta and fractional integrals. These inequalities extend some dynamic variants on time scales, and tie together and expand some integral inequalities. The present method is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractional differential equations applied in mathematical physics.

Stability and Existence Analysis to a Coupled System of Caputo Type Fractional Differential Equations with Erdelyi-Kober Integral Boundary Conditions

Stability and Existence Analysis to a Coupled System of Caputo Type Fractional Differential Equations with Erdelyi-Kober Integral Boundary Conditions

Tempered fractional Poisson processes and fractional equations with Z-transform

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the results available on fractional Poisson processes in several directions.

Multiple solutions for a class of p-Laplacian type fractional boundary value problems with instantaneous and non-instantaneous impulses

In this paper, we study the multiplicity of solutions for a class of p-Laplacian type fractional differential equations with both instantaneous and non-instantaneous impulses. By using the critical points theorem of B. Ricceri, a theorem is developed for the existence of at least three solutions for the impulsive problem depending on two parameters.

Impulsive boundary value problems for nonlinear implicit Caputo-exponential type fractional differential equations

This paper deals with existence and uniqueness of solutions to a class of impulsive boundary value problem for nonlinear implicit fractional differential equations involving the Caputo-exponential fractional derivative. The existence results are based on Schaefer’s fixed point theorem and the uniqueness result is established via Banach’s contraction principle. Two examples are given to illustrate the main results.

On the Stability of Duffing Type Fractional Differential Equation with Cubic Nonlinearity

The purpose of this paper is to study the stability of nonlinear fractional Duffing equation where , by analysing the eigenvalues generated from the system of the given differential equation. Graphical results furthermore show the effect of the damping a...

On Solvability of an Initial Value Problem for Hilfer type Fractional Differential Equation with Nonlinear Maxima

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima. It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration. It is proved also the stability of the desired solution with respect to given parameter.

Weakly absolutely continuous pseudo-differentiable solutions to some Hadamard-Caputo type fractional differential equation

Weakly absolutely continuous pseudo-differentiable solutions to some Hadamard-Caputo type fractional differential equation

On the new fractional hybrid boundary value problems with three-point integral hybrid conditions

We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results.

Free Oscillation Solution for Fractional Differential System

AbstractThere is a type of fractional differential equation that admits asymptotically free standing oscillations (Fukunaga, M., 2019, “Mode Analysis on Onset of Turing Instability in Time-Fractional Reaction-Subdiffusion Equations by Two-Dimensional Numerical Simulations,” ASME J. Comput. Nonlinear Dyn., 14, p. 061005). In this paper, analytical solutions to fractional differential equation for free oscillations are derived for special cases. These analytical solutions are direct evidence for asymptotically standing oscillations, while numerical solutions give indirect evidence.

Fractional differential equations of Sobolev type with sectorial operators

This paper treats the asymptotic behavior of resolvent operators of Sobolev type and its applications to the existence and uniqueness of mild solutions to fractional functional evolution equations of Sobolev type in Banach spaces. We first study the asymptotic decay of some resolvent operators (also called solution operators) and next, by using fixed point results, we obtain the existence and uniqueness of solutions to a class of Sobolev type fractional differential equation. We notice that, the existence or compactness of an operator $$E^{-1}$$ is not necessarily needed in our results.

Existence and multiplicity for some boundary value problems involving Caputo and Atangana–Baleanu fractional derivatives: A variational approach

In this paper, we study the existence and the numerical estimates of solutions for a specific types of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at the origin and infinity. We also take into consideration several theoretical and numerical examples. One of the main novelties of this paper is to use the variational method to investigate the properties of solutions of boundary values problems involving Atangana–Baleanu fractional derivative for the first time.

The Adomian Decomposition Method for a Type of Fractional Differential Equations

Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.

THE FRACTIONAL LAPLACE TRANSFORM SOLUTION FOR FRACTIONAL DIFFERENTIAL EQUATION THE OSCILLATOR IN THE PRESENCE OF AN EXTERNAL FORCES

The Fractional Laplace transform is applied to derive the solution of fractional differential equation of order governing the motion of the Harmonic Oscillator in a Non-Resisting Medium. The Fractional Laplacetransform method is to provide a potential tool to solve such a type of Fractional differential equation. Further consider oscillator in the presence of an external driving force of is in the fractional form and investigate some interesting features of the given equation. Also discuss the Numerical form of the oscillator in a r. The Fractional Laplace transform is applied to derive the solution of fractional differential equation of order governing the motion of the Harmonic Oscillator in a Non-Resisting Medium. The Fractional Laplacetransform method is to provide a potential tool to solve such a type of Fractional differential equation. Further consider oscillator in the presence of an external driving force of is in the fractional form and investigate some interesting features of the given equation. Also discuss the Numerical form of the oscillator in a resisting medium and discuss some cases of oscillation fractional differential equations for external driving force. esisting medium and discuss some cases of oscillation fractional differential equations for external driving force.

Existence results for Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Density-Dependent Conformable Space-time Fractional Diffusion-Reaction Equation and Its Exact Solutions

In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular ap-proximation techniques. In this research, we introduce spectral collocation method based on Lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1st kind give lowest accuracy in the proposed method.

Dynamics and stability of Hilfer-Hadamard type fractional differential equations with boundary conditions

In this paper, we investigate the existence, uniqueness and Ulam stabilities of solutions for Hilfer-Hadamard type fractional differential equations with boundary conditions in weighted spaces of continuous functions. The existence results rely on Schaefer's fixed point theorem. The Banach contraction principle is also considered to obtain uniqueness and stability results. An example is provided to illustrate the usefulness of the obtained results.