Duan-Rach Modified Decomposition Method for Solving Some Types of Nonlinear Fractional Multi-Point Boundary Value Problems

Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

“…e value of the positive integer N can be calculated using the homogeneous balance principle, i.e., by balancing between the highest order derivatives and the nonlinear terms appearing in (5). More precisely, if the degree of U(ξ) is Deg[U(ξ)] � N, then the degree of the other terms will be expressed as follows:…”

“…Step 2. We determine the positive integer N in (13) by applying the homogeneous balance principle to (5). e balancing rules are already described in (8) of the previous method.…”

“…Step 3. Substituting (13) along with ( 14) into (5) and collecting all of the coefficients of [ψ ′ (ξ)] q [ψ(ξ)] p , where q � 0, 1 and p � 0, ± 1, ± 2, . .…”

“…ere are two essential types of solutions for NPDEs, which are analytical and exact solutions. Some examples of the schemes used to obtain analytical approximate solutions to NPDEs are the homotopy analysis method (HAM) [3], the Adomian decomposition method (ADM) [4,5], the modified Laplace variational iteration method (ML-VIM) [6], and the reduced differential transform method [7], while many effective methods have been proposed to obtain exact solutions of NPDEs including fractional order partial differential equations such as the generalized Kudryashov method [8][9][10], the amplitude ansatz method [11], the He's semiinverse method [12,13], the exp-function method [14,15], the auxiliary equation method [16,17], the extended trial equation method [18,19], and the extended direct algebraic method [20]. Moreover, the sine-cosine method [21,22], the tanh-coth method [23,24], the extended sech-tanh method [25], the sine-Gordon expansion method [26][27][28], and the (G ′ /G)-expansion method [29][30][31] have been recently utilized to find analytical exact solutions of NPDEs as well.…”