Fractional variational iteration method and its application

The Combined Laplace-homotopy Analysis Method for Partial Differential Equations

“…Example 3.4. Let us consider the one dimensional non-homogeneous problem 27) subject to boundary conditions u(0, t) = sin t, u x (0, t) = 1 − sin t, and the initial condition u(x, 0) = x that is easily seen to have the exact solution u(x, t) = x+e −x sin t.…”

“…The Laplace transform is totally incapable of handling nonlinear equations because of the difficulties that are caused by the nonlinear terms. Various ways have been proposed recently to deal with these nonlinearities such as the Adomian decomposition method [23] and the Laplace decomposition algorithm [24,25,26,27,29]. Inspired and motivated by the ongoing research in this area, we use the homotopy analysis method coupled with the Laplace transformation for solving the linear and nonlinear equations in this paper.…”

An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

“…This fractional derivative was successfully implemented to fractional Laplace problems [35], fractional variational calculus [36], and probability calculus [37]. Jumarie's modified Riemann-Liouville derivative has many interesting properties: the˛order derivative of a constant is zero and it can be applied to both differentiable and nondifferentiable functions.…”

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

“…Moreover, the investigation of exact and approximate solutions for NFPDEs arising in mathematical physics, chemistry, biology, engineering, control theory, signal processing and so forth has become one of the most active and important research areas. A variety of analytical and numerical techniques have been well established and applied to solve NFPDEs, including the homogeneous balance method [6], the fractional sub-equation method [7][8][9][10][11], the exp-function method [12], the (G ′ /G)-expansion method [13,14], the first integral method [15], the modified trial equation method [16], the Jacobi elliptic equation method [17], the modified 49 Kudryashov method [18], the homotopy analysis transform method [19], the fractional variational iteration method [20], the Adomian decomposition method [21], and so on. In many analytical methods, the fractional complex transformation proposed by Li and He [22] plays a key role in converting NFPDEs into NODEs.…”