scholarly journals Rapid Convergence of Approximate Solutions for Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xiran Wu ◽  
Junyan Bao ◽  
Yufeng Sun
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Forcing Function ◽  
Fractional Differential ◽  
Convergence Of Approximate Solutions ◽  
Quasilinearization Technique

In this paper, we develop a generalized quasilinearization technique for a class of Caputo’s fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of order m (m≥0), and we obtain the convergence of the sequences of approximate solutions by establishing the convergence of order k (k≥2).

scholarly journals Generalized Quasilinearization for the System of Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Peiguang Wang ◽  
Ying Hou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
Monotone Sequences ◽  
Fractional Differential ◽  
Quasilinearization Technique

This paper considers the initial value problems of the system of fractional differential equations and constructs two monotone sequences of upper and lower solutions. By using quasilinearization technique, monotone sequences of approximate solutions that converge quadratically to a solution are obtained.

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.

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 Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yameng Wang ◽  
Juan Zhang ◽  
Yufeng Sun
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Lower And Upper Solutions ◽  
Quasilinearization Method ◽  
Rapid Convergence ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
Convergence Of Approximate Solutions

In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method.

scholarly journals An Iterative Method for Time-Fractional Swift-Hohenberg Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Wenjin Li ◽  
Yanni Pang
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Cauchy Problems ◽  
Initial Value ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

We study a type of iterative method and apply it to time-fractional Swift-Hohenberg equation with initial value. Using this iterative method, we obtain the approximate analytic solutions with numerical figures to initial value problems, which indicates that such iterative method is effective and simple in constructing approximate solutions to Cauchy problems of time-fractional differential equations.

On the operational solutions of fuzzy fractional differential equations

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Djurdjica Takači ◽  
Arpad Takači ◽  
Aleksandar Takači
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Software Package ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Fractional Differential ◽  
Definition Of ◽  
The Given ◽  
Fuzzy Fractional Differential Equations

AbstractFuzzy fractional differential equations with fuzzy coefficients are analyzed in the frame of Mikusiński operators. Systems of fuzzy operational algebraic equations are obtained, in view of the definition of fuzzy derivatives. Their exact and approximate solutions are constructed and their characters are analyzed, considering them as the corresponding solutions of the given problem. The described procedure of the construction of solutions is illustrated on an example and the obtained approximate solutions of the considered problems are visualized by using the GeoGebra software package.

scholarly journals Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 724
Author(s):  
Kateryna Marynets
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Approximation Technique ◽  
Fractional Boundary Value Problem ◽  
Point Boundary ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

We studied one essentially nonlinear two–point boundary value problem for a system of fractional differential equations. An original parametrization technique and a dichotomy-type approach led to investigation of solutions of two “model”-type fractional boundary value problems, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called “bifurcation” equations.

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