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The purpose of this paper is to implement the variational iteration method and the homotopy perturbation method to give a rational approximation solution of the foam drainage equation with time‐ and space‐fractional derivatives.

The fractional derivatives are described in the Caputo sense. In these schemes, the solution takes the form of a convergent series with easily computable components.

Numerical examples are given to demonstrate the effectiveness of the present methods.

Results show that the proposed schemes are very effective and convenient for solving linear and nonlinear fractional differential equations with high accuracy.

The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.

In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.

We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.

Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.

This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.

This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The purpose of this paper is to consider the time‐fractional diffusion‐wave equation. The time‐fractional diffusion equation is obtained from the standard diffusion equation by replacing the first‐order time derivative with a fractional derivative of order α ∈ (0, 2]. The fractional derivatives are described in the Caputo sense.

The two methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations.

Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

In this paper, the variational iteration and homotopy perturbation methods are used to obtain a solution of a fractional diffusion equation.

The purpose of this paper is to apply a new modified homotopy perturbation method, which is effective to solve multi-order fractional equations with non-polynomial initial and boundary conditions.

The proposed algorithm is tested on multi-order fractional advection-dispersion equations. The fractional derivatives described in this paper are in the Caputo sense.

Approximate results explicitly reveal the complete reliability, efficiency and accuracy of the new modified technique.

It is observed that the approach may be implemented to other multi-fractional models with non-polynomial initial and boundary conditions.

The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters.

To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models.

The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail.

From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.

The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.

The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.

New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.

The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.

The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions.

The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method.

Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem.

This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.