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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.

This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.

The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.

This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.

The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.

This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.

The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.

This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.

The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion KdV (KdV8) and many other related equations.

The newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria.

The authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct resonance branches, whereas the phase shift retains the KdV type of shifts for any member.

This paper presents an efficient algorithm for handling a hierarchy of integrable equations of diverse orders.

Multisoliton solutions are derived for each member of the hierarchy, and then generalized for any higher-order model.

This work presents useful algorithms for finding and studying integrable equations of a hierarchy of nonlinear equations. The developed models exhibit complete integrability, by investigating the compatibility conditions for each model.

This paper presents an original work with a variety of useful findings.

The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.

The newly developed Painlevé integrable equations have been handled by using Hirota’s direct method. The authors obtain multiple soliton solutions and other kinds of solutions for these six models.

The developed Hamiltonian models exhibit complete integrability in analogy with the original equation.

The present study is to address these two main motivations: the study of the integrability features and solitons and other useful solutions for the developed equations.

The work introduces six Painlevé-integrable equations developed from a Hamiltonian model.

The work presents useful algorithms for constructing new integrable equations and for handling these equations.

The paper presents an original work with newly developed integrable equations and shows useful findings.

The purpose of this paper is to provide a Monte Carlo variance reduction method based on Control variates to solve Fredholm integral equations of the second kind.

A numerical algorithm consisted of the combined use of the successive substitution method and Monte Carlo simulation is established for the solution of Fredholm integral equations of the second kind.

Owing to the application of the present method, the variance of the solution is reduced. Therefore, this method achieves several orders of magnitude improvement in accuracy over the conventional Monte Carlo method.

Numerical tests are performed in order to show the efficiency and accuracy of the present paper. Numerical experiments show that an excellent estimation on the solution can be obtained within a couple of minutes CPU time at Pentium IV‐2.4 GHz PC.

This paper provides a new efficient method to solve Fredholm integral equations of the second kind and discusses basic advantages of the present method.

To show a possible implementation of surface impedance boundary conditions (SIBCs) in a time domain formulation based on the cell method (CM).

The implementation is based on vector fitting (VF), a technique which allows a time domain representation of a rational approximation of the surface impedance to be found.

It is shown that very little computational effort is needed to find a very good VF approximation of simple SIBCs and that such approximation is easily fitted into existing CM codes.

The extension to higher order SIBCs has not been taken into account.

The proposed approach avoids the use of convolution integrals, is accurate and easy to implement.

This paper introduces the use of VF for the approximate time domain representation of SIBCs.

The purpose of this paper is to investigate the three-dimensional flow of viscous fluid with convective boundary conditions and heat generation/absorption.

The governing partial differential equations are reduced into ordinary differential equations by applying similarity transformations. Series solutions of velocity and temperature are found by adopting homotopy analysis method (HAM).

The authors found that an increase in ratio parameter and Hartman number increased the values of skin-friction coefficient, but the values of local Nusselt number are reduced with an increase in Hartman number.

The present study is a useful source of information for the investigators in the fields of Newtonian fluids and heat transfer. The results obtained are specifically important in processes of polymer industry and metallurgical.

Very scarce literature is available on three-dimensional stretched flow with convective boundary conditions. Mathematical modelling for such flow in regime of magnetohydrodynamics is a main concern here. Performed computations to the resulting nonlinear analysis further enhance its novelty.

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its properties.

This disaggregation method includes three main steps. First, by utilizing Gamma function expanded for integer factorial, this paper expands one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, and gives the analytical expression of fractional order reducing generation operator. Then, studies the commutative law and exponential law of fractional order reducing generation operator. Lastly, gives several examples of fractional order reducing generation operator and verifies the commutative law and exponential law of fractional order reducing generation operator.

The authors pull the analytical expression of fractional order reducing generation operator and verify that fractional order reducing generation operator satisfies commutative law and exponential law.

Expanding the reducing generation operator would help develop grey prediction model with fractional order operators and widen the application fields of grey prediction models.

The analytical expression of fractional order reducing generation operator, properties of commutative law and exponential law for fractional order reducing generation operator are first studied.

To study the multiplicative perturbation of local C‐regularized cosine functions associated with the following incomplete second order abstract differential equations in a Banach space X u″(t)=A(I+B)u(t), u(0)=x, u′(0)=y,(*) where A is a closed linear operator on X and B is a bounded linear operator on X.

The multiplicative perturbation of exponentially bounded regularized C‐cosine functions is generally studied by the Laplace transformation. However, C‐cosine functions might not be exponentially bounded, so that the new method for the multiplicative perturbation of the nonexponentially bounded regularized C‐cosine functions should be applied. In this paper, the property of regularized C‐cosine functions is directly used to obtain the desired results.

The new results of the multiplicative perturbations of the nonexponentially bounded C‐cosine functions are obtained.

The new techniques differing from those given previously in the literature are employed to deduce the desired conclusions. The results can be applied to deal with incomplete second order abstract differential equations which stem from cybernetics, engineering, physics, etc.

The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).

This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative.

This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense.

It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.