This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions.

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform (FCT). This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method (HPM) is employed to tackle the nonlinear terms in the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two‐scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering.

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method ($\mathbb{S}$T-RPSM). This scheme uses a combination of the Shehu transform ($\mathbb{S}$T) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare $\mathbb{S}$T-RPSM results with exact solutions.

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity.

Wave problem arises in various phenomena of science and engineering. This study proposes an efficient and appropriate scheme to produce the approximate solutions of multidimensional problems arising in wave propagation. We use a new iterative strategy (NIS) to minimize the computational cost and less time than other approaches studied in the literature. Since the variational iteration method (VIM) has some assumptions and restrictions of variables during the construction of the recurrence relation, we present a Sawi integral transform for the construction of the recurrence to overcome this difficulty. The series solution for this recurrence relation is determined using the homotopy perturbation strategy (HPS) in the form of convergence that provides the precise solution after a few iterations. This NIS does not require any discretization in the findings of results and derives the algebraic equations. Some numerical models and graphical representations are illustrated to verify the performance of this scheme.