The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov’s fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for the proposed system. The approximation of the solution is given, and as a novelty, the approximation of its derivative is also obtained using the same iteration steps.
In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differential operator. We present two properties of this operator from which we constructed the final results. Moreover, based on the obtained outcomes, we give two sandwich-type theorems. Some interesting further consequences are also taken into consideration.
In this paper, new criteria for a class oscillation of second-order delay differential equations with distributed deviating arguments were established. Our method mainly depends on making sharper estimates for the non-oscillatory solutions of the studied equation. By using the Ricati technique and comparison theorems that compare the studied equations with first-order delay differential equations, we obtained new and less restrictive conditions that ensure the oscillation of all solutions of the studied equation. Further, we give an illustrative example.
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili (KP) model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation (DT). The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the DT is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed analytical techniques are useful and have not yet been explored in the context of the analyzed models. The proposed methodologies are valuable and can be applied to additional nonlinear evolutionary models employed to describe nonlinear physical models within the domain of nonlinear science.
In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.
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