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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.

This paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.

The authors formally use the simplified Hirota's method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.

The Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level, but integrability characteristics is justified through the Lax sense.

Multiple-soliton solutions are explored using the Hirota's bilinear method. The authors also furnish a class of lump solutions using distinct values of the parameters via the positive quadratic function method.

The authors also retrieve a bunch of other solutions of distinct structures such as solitonic, periodic solutions and ratio of trigonometric functions solutions.

This work formally furnishes algorithms for extending integrable equations and for the determination of lump solutions.

To the best of the authors’ knowledge, this paper introduces an original work with newly developed Lax-integrable equation and shows new useful findings.

The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.

By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.

The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.

The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

This paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.

Hirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.

The fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.

This paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.

The aim of this study is to offer a contemporary approach for getting optical soliton and traveling wave solutions for the Date–Jimbo–Kashiwara–Miwa equation.

The approach is based on a recently constructed ansätze strategy. This method is an alternative to the Painleve test analysis, producing results similarly, but in a more practical, straightforward manner.

The approach proved the existence of both singular and optical soliton solutions. The method and its application show how much better and simpler this new strategy is than current ones. The most significant benefit is that it may be used to solve a wide range of partial differential equations that are encountered in practical applications.

The approach has been developed recently, and this is the first time that this method is applied successfully to extract soliton solutions to the Date–Jimbo–Kashiwara–Miwa equation.

This study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.

We use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.

Various conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.

This is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

Pores and shrink holes are unavoidable defects in the die-casting mass production process which may significantly influence the strength, fatigue and fracture behaviour as well as the life span of structures, especially if they are subjected to high static and dynamic loads. Such defects should be considered during the design process or after production, where the defects could be detected with the help of computed tomography (CT) measurements. However, this is usually not done in today's mass production environments. This paper deals with the stress analysis of die-cast structural parts with pores found from CT measurements or that are artificially placed within a structure.

In this paper the authors illustrate two general methodologies to take into account the porosity of die-cast components in the stress analysis. The detailed geometry of a die-cast part including all discontinuities such as pores and shrink holes can be included via STL data provided by CT measurements. The first approach is a combination of the finite element method (FEM) and the finite cell method (FCM), which extends the FEM if the real geometry cuts finite elements. The FCM is only applied in regions with pores. This procedure has the advantage that all simulations with different pore distributions, real or artificial, can be calculated without changing the base finite element mesh. The second approach includes the pore information as STL data into the original CAD model and creates a new adapted finite element mesh for the simulation. Both methods are compared and evaluated for an industrial problem.

The STL data of defects which the authors received from CT measurements could not be directly applied without repairing them. Therefore, for FEM applications an appropriate repair procedure is proposed. The first approach, which combines the FEM with the FCM, the authors have realized within the commercial software tool Abaqus. This combination performs well, which is demonstrated for test examples, and is also applied for a complex industrial project. The developed in-house code still has some limitations which restrict broader application in industry. The second pure FEM-based approach works well without limitations but requires increasing computational effort if many different pore distributions are to be investigated.

A new simulation approach which combines the FEM with the FCM has been developed and implemented into the commercial Abaqus FEM software. This approach the authors have applied to simulate a real engineering die-cast structure with pores. This approach could become a preferred way to consider pores in practical applications, where the porosity can be derived either from CT measurements or are artificially adopted for design purposes. The authors have also shown how pores can be considered in the standard FEM analysis as well.

In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.

We introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.

We develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.

We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC. In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.