scholarly journals Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shaochun Ji ◽  
Dandan Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Integrodifferential System ◽  
Liouville Fractional Derivative

AbstractThis paper is concerned with the semilinear fractional integrodifferential system with Riemann–Liouville fractional derivative. Firstly, we introduce the suitable $C_{1-\alpha }$C1−α-solution to Riemann–Liouville fractional integrodifferential equations in the new frame of fractional resolvents. Some properties of fractional resolvents are given. Then we discuss the sufficient conditions for the existence of solutions without the Lipschitz assumptions to nonlinear item. Finally, an example on fractional partial differential equations is presented to illustrate our abstract results.

scholarly journals Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Wei Nian Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Forced Oscillation ◽  
Smooth Boundary ◽  
Sufficient Conditions ◽  
Normal Vector ◽  
Fractional Partial Differential Equations ◽  
Damping Term ◽  
Partial Differential ◽  
Liouville Fractional Derivative

Sufficient conditions are established for the forced oscillation of fractional partial differential equations with damping term of the form(∂/∂t)(D+,tαu(x,t))+p(t)D+,tαu(x,t)=a(t)Δu(x,t)-q(x,t)u(x,t)+f(x,t),(x,t)∈Ω×R+≡G, with one of the two following boundary conditions:∂u(x,t)/∂N=ψ(x,t),  (x,t)∈∂Ω×R+oru(x,t)=0,  (x,t)∈∂Ω×R+, whereΩis a bounded domain inRnwith a piecewise smooth boundary,∂Ω,R+=[0,∞),  α∈(0,1)is a constant,D+,tαu(x,t)is the Riemann-Liouville fractional derivative of orderαofuwith respect tot,Δis the Laplacian inRn,Nis the unit exterior normal vector to∂Ω, andψ(x,t)is a continuous function on∂Ω×R+. The main results are illustrated by some examples.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

2021 ◽  
pp. 2150492
Author(s):  
Delmar Sherriffe ◽  
Diptiranjan Behera ◽  
P. Nagarani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Function Method ◽  
Visual Representations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Varying Parameters ◽  
Wave Solutions ◽  
Soliton Wave Solutions

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

scholarly journals The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 771
Author(s):  
Yusuf Gurefe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method

In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

scholarly journals A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

2021 ◽  
Vol 9 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Zeyad Al-Zhour
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Hyperbolic System ◽  
Series Solution ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

scholarly journals Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xianghu Liu ◽  
Yanfang Li
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Existence Of Solutions ◽  
Integral Formula ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Uniqueness Of Solutions ◽  
Liouville Fractional Derivative

This paper is concerned with the sufficient conditions for the existence of solutions for a class of generalized antiperiodic boundary value problem for nonlinear fractional impulsive differential equations involving the Riemann-Liouville fractional derivative. Firstly, we introduce the fractional calculus and give the generalized R-L fractional integral formula of R-L fractional derivative involving impulsive. Secondly, the sufficient condition for the existence and uniqueness of solutions is presented. Finally, we give some examples to illustrate our main results.

scholarly journals Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ai-Min Yang ◽  
Jie Li ◽  
H. M. Srivastava ◽  
Gong-Nan Xie ◽  
Xiao-Jun Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Variational Iteration ◽  
Linear Partial Differential Equations

The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

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