A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense ofmodified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled...

This work is devoted to the study of global solution for initial value problem of interval fractional integrodifferential equations involving Caputo-Fabrizio fractional derivative without singular kernel admitting only the existence of a lower solution or an upper solution. Our method is based on fixed point in partially ordered sets. In this study, we guaranty the existence of special kind of ...

In this article, Ritz approximation have been employed to obtain the numerical solutions of a class of the fractional optimal control problems based on the Caputo fractional derivative. Using polynomial basis functions, we obtain a system of nonlinear algebraic equations. This nonlinear system of equation is solved and the coefficients of basis polynomial are derived. The convergence of the num...

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative...

and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving diffi...

This paper deals with the application of fourth kind Chebyshev wavelets (FKCW) in solving numerically a model of HIV infection of CD4+T cells involving Caputo fractional derivative. The present problem is a system of nonlinear fractional differential equations. The goal is to approximate the solution in the form of FKCW truncated series. To do this, an operational matrix of fractional integrati...

and Applied Analysis 3 Definition 2.2. The Riemann-Liouville fractional derivative of order α > 0 of a function f : 0,∞ → R is given by D 0 f t 1 Γ n − α ( d dt )n ∫ t 0 f s t − s α−n 1 ds, 2.2 where n α 1 and α denotes the integer part of α. The following two lemmas can be found in 17, 22 . Lemma 2.3. Let α > 0 and u ∈ C 0, 1 ∩ L1 0, 1 . Then fractional differential equation D 0 u t 0 2.3

In this paper, we study the existence of positive solutions to boundary value problem for fractional differential equation  cDσ 0+(φp(u ′′(t)))− g(t)f(u(t)) = 0, t ∈ (0, 1), φp(u ′′(1)) = 0, φp(u ′′(0)) = 0, αu(0)− βu′(0) = 0, γu(1) + δu′(1) = 0, where cDα 0+ is the Caputo’s fractional derivative of order 1 < σ ≤ 2, φp(s) = |s| p−2s, p > 1, α, β, γ, δ ≥ 0 and f ∈ C([0,∞); [0,∞)), g ∈ C((...

in this paper, exp-function and (g′/g)expansion methods are presented to derive traveling wave solutions for a class of nonlinear space-time fractional differential equations. as a results, some new exact traveling wave solutions are obtained.

and Applied Analysis 3 where n α 1, α denotes the integer part of number α, provided that the right side is pointwise defined on 0, ∞ . Definition 2.2 see 20 . The Riemann-Liouville fractional integral of order α > 0 of a function f : 0, ∞ → R is given by I 0 f t 1 Γ α ∫ t 0 t − s α−1f s ds, 2.2 provided that the right side is pointwise defined on 0, ∞ . From the definition of the Riemann-Liouv...