scholarly journals On fractional KdV-burgers and potential KdV equations: Existence and uniqueness results

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2107-2117
Author(s):  
Mir Hashemi ◽  
Mustafa Inc ◽  
Dumitru Baleanu
Keyword(s):  
Exact Solutions ◽  
Invariant Subspace ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Conformable Derivative ◽  
Existence And Uniqueness Results ◽  
Kdv Equations ◽  
Uniqueness Results ◽  
Conformable Derivatives

Recently a new kind of derivatives, namely the conformable derivative is introduced which have not many drawbacks of other fractional derivatives. Two types of KdV equations with conformable derivative are investigated in this paper. Ex-istence and uniqueness of two different equations of KdV class with conformable derivatives are investigated. It is also shown that the invariant subspace method can be extended to find the exact solutions of these equations.

scholarly journals A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 35-48 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Partial Differential ◽  
Singular Kernels ◽  
Fractional Differential

AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

scholarly journals Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Caputo Fractional Derivatives ◽  
Existence And Uniqueness Results ◽  
Intuitionistic Fuzzy ◽  
Uniqueness Results ◽  
Fractional Boundary Value Problems

In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.

Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zohre Kiyamehr ◽  
Hamid Baghani
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Langevin Equations ◽  
Point Boundary ◽  
Caputo Fractional Derivatives ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

AbstractThis article investigates a nonlinear fractional Caputo–Langevin equationD^{\beta}(D^{\alpha}+\lambda)x(t)=f(t,x(t)),\quad 0<t<1,\,0<\alpha\leq 1,\,1<% \beta\leq 2,subject to the multi-point boundary conditionsx(0)=0,\qquad\mathcal{D}^{2\alpha}x(1)+\lambda\mathcal{D^{\alpha}}x(1)=0,% \qquad x(1)=\int_{0}^{\eta}x(\tau)\,d\tau\quad\text{for some }0<\eta<1,where {D^{\alpha}} is the Caputo fractional derivative of order α, {f:[0,1]\times\mathbb{R}\to\mathbb{R}} is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

scholarly journals Existence and uniqueness results for fractional Navier boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imed Bachar ◽  
Hassan Eltayeb
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Real Function ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Alpha Beta

Abstract We establish the existence, uniqueness, and positivity for the fractional Navier boundary value problem: $$\begin{aligned} \textstyle\begin{cases} D^{\alpha }(D^{\beta }\omega )(t)=h(t,\omega (t),D^{\beta }\omega (t)), & 0< t< 1, \\ \omega (0)=\omega (1)=D^{\beta }\omega (0)=D^{\beta }\omega (1)=0, \end{cases}\displaystyle \end{aligned}$$ { D α ( D β ω ) ( t ) = h ( t , ω ( t ) , D β ω ( t ) ) , 0 < t < 1 , ω ( 0 ) = ω ( 1 ) = D β ω ( 0 ) = D β ω ( 1 ) = 0 , where $\alpha,\beta \in (1,2]$ α , β ∈ ( 1 , 2 ] , $D^{\alpha }$ D α and $D^{\beta }$ D β are the Riemann–Liouville fractional derivatives. The nonlinear real function h is supposed to be continuous on $[0,1]\times \mathbb{R\times R}$ [ 0 , 1 ] × R × R and satisfy appropriate conditions. Our approach consists in reducing the problem to an operator equation and then applying known results. We provide an approximation of the solution. Our results extend those obtained in (Dang et al. in Numer. Algorithms 76(2):427–439, 2017) to the fractional setting.

scholarly journals Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2543
Author(s):  
Abdulkafi M. Saeed ◽  
Mohammed S. Abdo ◽  
Mdi Begum Jeelani
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Special Cases ◽  
Stability Of The Solution

In this research paper, we consider a class of a coupled system of fractional integrodifferential equations in the frame of Hilfer fractional derivatives with respect to another function. The existence and uniqueness results are obtained in weighted spaces by applying Schauder’s and Banach’s fixed point theorems. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam–Hyers stability of the solution to the proposed system. Some examples are also constructed to illustrate and validate the main results.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

Nontrivial solutions of non-autonomous dirichlet fractional discrete problems

2020 ◽  
Vol 23 (4) ◽  
pp. 980-995
Author(s):  
Alberto Cabada ◽  
Nikolay Dimitrov
Keyword(s):  
Existence And Uniqueness ◽  
Point Boundary ◽  
Nontrivial Solutions ◽  
Fractional Difference ◽  
Nonlinear Part ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Difference Equation ◽  
Discrete Problems ◽  
Series Of Functions

AbstractIn this paper, we introduce a two-point boundary value problem for a finite fractional difference equation with a perturbation term. By applying spectral theory, an associated Green’s function is constructed as a series of functions and some of its properties are obtained. Under suitable conditions on the nonlinear part of the equation, some existence and uniqueness results are deduced.

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