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In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.

For, the authors have used the notion of conformal transformation and Douglas space.

The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.

The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations.

The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model.

The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow.

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions.

The work presents useful findings in the propagation of waves.

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

A generalization of Ascoli–Arzelá theorem in Banach spaces is established. Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order. The authors’ results are obtained under, rather, general assumptions.

First, a generalization of Ascoli–Arzelá theorem in Banach spaces in Cⁿ is established. Second, this new generalization with Schauder's fixed point theorem to prove the existence of a solution for a boundary value problem of higher order is used. Finally, an illustrated example is given.

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In this work, a new generalization of Ascoli–Arzelá theorem in Banach spaces in Cⁿ is established. To the best of the authors’ knowledge, Ascoli–Arzelá theorem is given only in Banach spaces of continuous functions. In the second part, this new generalization with Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order, where the derivatives appear in the non-linear terms.

The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models.

The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.

The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions.

The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients.

This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions.

The paper presents useful algorithms for finding integrable equations with time-dependent coefficients.

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

In systems theory, authors such as Klir, Miller, Yang, Lin and Ma, Backlund, etc. have developed different definitions of the system concept in function of both the type of variables used and the type of connection between variables. The concept of the subsystem, however, tends not to vary substantially from author to author, and this leads to a new system definition based on the subsystem concept, analysing the possible cases of interaction between subsystems and obtaining results for the overall system from an analysis of its subsystems.

This paper derives three evaluation measures of file recovery with n archive copies: when a failure is detected uniformly during (0, T),the mean recovery overhead, the overhead rate and the availability are obtained. Optimal archive copies intervals T* which minimize or maximize those measures are analytically discussed. When faults occur at random, they are given by unique solutions of the same type equations. Finally, a numerical example is shown.

The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation.

Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.

Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.