scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

scholarly journals Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ya-ling Li ◽  
Shi-you Lin
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Continuous Function ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fixed Point Theorems ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

scholarly journals Oscillation for a Class of Fractional Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Zhenlai Han ◽  
Yige Zhao ◽  
Ying Sun ◽  
Chao Zhang
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Real Number ◽  
Fractional Differential Equation ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Nonlinear Fractional Differential Equation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Fractional Differential

We consider the oscillation for a class of fractional differential equation[r(t)g(D-αy)(t)]'-p(t)f∫t∞‍(s-t)-αy(s)ds=0,fort>0,where0<α<1is a real number andD-αyis the Liouville right-sided fractional derivative of orderαofy. By generalized Riccati transformation technique, oscillation criteria for a class of nonlinear fractional differential equation are obtained.

Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions

2018 ◽  
Vol 21 (3) ◽  
pp. 833-843 ◽  
Author(s):  
Youyu Wang ◽  
Qichao Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Point Boundary ◽  
Hilfer Fractional Derivative ◽  
Nonlinear Fractional Differential Equation ◽  
Early Results ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

Abstract In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Hilfer fractional derivative under multi-point boundary conditions, the results are new and generalize and improve some early results in the literature.

scholarly journals Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

2020 ◽  
Vol 17 (4) ◽  
pp. 1234
Author(s):  
Nour Salman ◽  
Muna Mansour Mustfaf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Fractional Differential Equation ◽  
Lagrange Polynomial ◽  
Lagrange Polynomials ◽  
Fractional Differential ◽  
A New Technique ◽  
Special Case

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems.

scholarly journals Analytical solutions for conformable fractional Bratu-type equations

2018 ◽  
Vol 7 (1) ◽  
pp. 15 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Conformable Fractional Derivative ◽  
Fractional Differential

Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.

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