Approximate Solutions to Nonlinear Fractional Differential Equations with the Caputo-Fabrizio Derivative

2020 ◽  
Vol 6 (4) ◽  
pp. 301-312
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Approximate Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Yiliang Liu ◽  
Liang Lu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Nonlinear Boundary Conditions ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

AbstractIn this paper, we deal with multiple solutions of fractional differential equations with p-Laplacian operator and nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions. An example is given to illustrate our results.

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