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This paper presents a Monotonic Unbounded Schemes Transformer (MUST) approach to bound/monotonize (remove undershoots and overshoots) unbounded spatial differencing schemes automatically, and naturally. Automatically means the approach (1) captures the critical cell Peclet number when an unbounded scheme starts to produce physically unrealistic solution automatically, and (2) removes the undershoots and overshoots as part of the formulation without requiring human interventions. Naturally implies, all the terms in the discretization equation of the unbounded spatial differencing scheme are retained.

The authors do not formulate new higher-order scheme. MUST transforms an unbounded higher-order scheme into a bounded higher-order scheme.

The solutions obtained with MUST are identical to those without MUST when the cell Peclet number is smaller than the critical cell Peclet number. For cell Peclet numbers larger than the critical cell Peclet numbers, MUST sets the nodal values to the limiter value which can be derived for the problem at-hand. The authors propose a way to derive the limiter value. The authors tested MUST on the central differencing scheme, the second-order upwind differencing scheme and the QUICK differencing scheme. In all cases tested, MUST is able to (1) capture the critical cell Peclet numbers; the exact locations when overshoots and undershoots occur, and (2) limit the nodal value to the value of the limiter values. These are achieved by retaining all the discretization terms of the respective differencing schemes naturally and accomplished automatically as part of the discretization process. The authors demonstrated MUST using one-dimensional problems. Results for a two-dimensional convection–diffusion problem are shown in Appendix to show generality of MUST.

The authors present an original approach to convert any unbounded scheme to bounded scheme while retaining all the terms in the original discretization equation.

The paper aims to study the constraint solutions of the periodic coupled operator matrix equations by the biconjugate residual algorithm. The new algorithm can solve a lot of constraint solutions including Hamiltonian solutions and symmetric solutions, as special cases. At the end of this paper, the new algorithm is applied to the pole assignment problem.

When the studied periodic coupled operator matrix equations are consistent, it is proved that constraint solutions can converge to exact solutions. It is demonstrated that the solutions of the equations can be obtained by the new algorithm with any arbitrary initial matrices without rounding error in a finite number of iterative steps. In addition, the least norm-constrained solutions can also be calculated by selecting any initial matrices when the equations of the periodic coupled operator matrix are inconsistent.

Numerical examples show that compared with some existing algorithms, the proposed method has higher convergence efficiency because less data are used in each iteration and the data is sufficient to complete an update. It not only has the best convergence accuracy but also requires the least running time for iteration, which greatly saves memory space.

Compared with previous algorithms, the main feature of this algorithm is that it can synthesize these equations together to get a coupled operator matrix equation. Although the equation of this paper contains multiple submatrix equations, the algorithm in this paper only needs to use the information of one submatrix equation in the equation of this paper in each iteration so that different constraint solutions of different (coupled) matrix equations can be studied for this class of equations. However, previous articles need to iterate on a specific constraint solution of a matrix equation separately.

This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates the connections between the differential geometry of curves and the concept of equivalence.

The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability.

This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion relations and phase shifts.

Hirota’s bilinear technique is used to provide these novel solutions.

This study also provides a diverse range of solutions for the K-II-X equation, including kink, periodic and singular solutions.

This study provides formal procedures for analyzing recently developed systems that investigate optical communications, plasma physics, oceans and seas, fluid mechanics and the differential geometry of curves, among other topics.

The study introduces a novel Painlevé integrable model that has been constructed and delivers valuable discoveries.

The boundary integral method (BIM) is very attractive to practicing engineers as it reduces the dimensionality of the problem by one, thereby making the procedure computationally inexpensive compared to its peers. The principal feature of this technique is the limitation of all its computations to only the boundaries of the domain. Although the procedure is well developed for the Laplace equation, the Poisson equation offers some computational challenges. Nevertheless, the literature provides a couple of solution methods. This paper revisits an alternate approach that has not gained much traction within the community. The purpose of this paper is to address the main bottleneck of that approach in an effort to popularize it and critically evaluate the errors introduced into the solution by that method.

The primary intent in the paper is to work on the particular solution of the Poisson equation by representing the source term through a Fourier series. The evaluation of the Fourier coefficients requires a rectangular domain even though the original domain can be of any arbitrary shape. The boundary conditions for the homogeneous solution gets modified by the projection of the particular solution on the original boundaries. The paper also develops a new Gauss quadrature procedure to compute the integrals appearing in the Fourier coefficients in case they cannot be analytically evaluated.

The current endeavor has developed two different representations of the source terms. A comprehensive set of benchmark exercises has successfully demonstrated the effectiveness of both the methods, especially the second one. A subsequent detailed analysis has identified the errors emanating from an inadequate number of boundary nodes and Fourier modes, a high difference in sizes between the particular solution and the original domains and the used Gauss quadrature integration procedures. Adequate mitigation procedures were successful in suppressing each of the above errors and in improving the solution accuracy to any desired level. A comparative study with the finite difference method revealed that the BIM was as accurate as the FDM but was computationally more efficient for problems of real-life scale. A later exercise minutely analyzed the heat transfer physics for a fin after validating the simulation results with the analytical solution that was separately derived. The final set of simulations demonstrated the applicability of the method to complicated geometries.

First, the newly developed Gauss quadrature integration procedure can efficiently compute the integrals during evaluation of the Fourier coefficients; the current literature lacks such a tool, thereby deterring researchers to adopt this category of methods. Second, to the best of the author’s knowledge, such a comprehensive error analysis of the solution method within the BIM framework for the Poisson equation does not currently exist in the literature. This particular exercise should go a long way in increasing the confidence of the research community to venture into this category of methods for the solution of the Poisson equation.

The purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov’s methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation.

The article’s methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions.

As for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation.

The originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field.

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.