A solution to a fractional order semilinear equation using variational method

2020 ◽  
Vol 99 (99) ◽  
pp. 1-11
Author(s):  
Ramesh Karki
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Fractional Order ◽  
Direct Method ◽  
Lagrange Equation ◽  
Fractional Power ◽  
Semilinear Equation ◽  
Pseudo Differential Equation ◽  
Energy Type ◽  
The Given

We will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of laplacian by using a method analogous to the direct method of calculus of variations. More precisely, we will discuss the existence of a minimizer of a suitable energy-type functional whose Euler-Lagrange equation is the given semilinear pseudo-differential equation, and also discuss the regularity of such a minimizer so that it will be a solution to the semilinear equation.

scholarly journals Travelling waves solution for fractional-order biological population model

2021 ◽  
Vol 16 ◽  
pp. 32
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
J.F. Gómez-Aguilar ◽  
Shoaib ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Traveling Waves ◽  
Population Model ◽  
Travelling Waves ◽  
Order Problem ◽  
Physical Behavior ◽  
Biological Population ◽  
The Given

In this paper, we implemented the generalized (G′/G) and extended (G′/G) methods to solve fractional-order biological population models. The fractional-order derivatives are represented by the Caputo operator. The solutions of some illustrative examples are presented to show the validity of the proposed method. First, the transformation is used to reduce the given problem into ordinary differential equations. The ordinary differential equation is than solve by using modified (G′/G) method. Different families of traveling waves solutions are constructed to explain the different physical behavior of the targeted problems. Three important solutions, hyperbolic, rational and periodic, are investigated by using the proposed techniques. The obtained solutions within different classes have provided effective information about the targeted physical procedures. In conclusion, the present techniques are considered the best tools to analyze different families of solutions for any fractional-order problem.

Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Boonrod Yuttanan ◽  
Mohsen Razzaghi ◽  
Thieu N. Vo
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Algebraic System ◽  
Fractional Integral ◽  
Exact Formula ◽  
Legendre Wavelets ◽  
Riccati Differential Equation ◽  
Numerical Examples ◽  
Riccati Differential Equations ◽  
The Given

Abstract In the present paper, fractional-order generalized Legendre wavelets (FOGLWs) are introduced. We apply the FOGLWs for solving fractional Riccati differential equation. By using the hypergeometric function, we obtain an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FOGLWs. By using this exact formula and the properties of the FOGLWs, we reduce the solution of the fractional Riccati differential equation to the solution of an algebraic system. This algebraic system can be solved effectively. This method gives very accurate results. The given numerical examples support this claim.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

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