Advances in Continuous and Discrete Models

In this paper, the dynamical behavior of a mathematical model of cancer including tumor cells, immune cells, and normal cells is investigated when a delay term is induced. Though the model was originally proposed by De Pillis et al. (Math. Comput. Model. 37:1221–1244, 2003), to make the model more realistic, we have added a delay term into the model, and it has incorporated novelty in our present work. The stability of existing equilibrium points in the delay-induced system is studied in detail. Global stability conditions of the tumor-free equilibrium point have been found. It is shown that due to this delay effect, the coexisting equilibrium point may lose its stability through a Hopf bifurcation. The implicit function theorem is applied to characterize a complex function in a neighborhood of delay terms. Additionally, the presence of Hopf bifurcation is demonstrated when the transversality conditions are satisfied. The length of delay for which the solutions preserve the stability of the limit cycle is estimated. Finally, through a series of numerical simulations, the theoretical results are formally examined.

According to the theory of regular geometric functions, the relevance of geometry to analysis is a critical feature. One of the significant tools to study operators is to utilize the convolution product. The dynamic techniques of convolution have attracted numerous complex analyses in current research. In this effort, an attempt is made by utilizing the said techniques to study a new linear complex operator connecting an incomplete beta function and a Hurwitz–Lerch zeta function of certain meromorphic functions. Furthermore, we employ a method based on the first-order differential subordination to derive new and better differential complex inequalities, namely differential subordinations.

Study of ecosystems has always been an interesting topic in the view of real-world dynamics. In this paper, we propose a fractional-order nonlinear mathematical model to describe the prelude of deteriorating quality of water cause of greenhouse gases on the population of aquatic animals. In the proposed system, we recall that greenhouse gases raise the temperature of water, and because of this reason, the dissolved oxygen level goes down, and also the rate of circulation of disintegrated oxygen by the aquatic animals rises, which causes a decrement in the density of aquatic species. We use a generalized form of the Caputo fractional derivative to describe the dynamics of the proposed problem. We also investigate equilibrium points of the given fractional-order model and discuss the asymptotic stability of the equilibria of the proposed autonomous model. We recall some important results to prove the existence of a unique solution of the model. For finding the numerical solution of the established fractional-order system, we apply a generalized predictor–corrector technique in the sense of proposed derivative and also justify the stability of the method. To express the novelty of the simulated results, we perform a number of graphs at various fractional-order cases. The given study is fully novel and useful for understanding the proposed real-world phenomena.

Trong bài báo này, chúng tôi phân tích một bộ tích phân mới theo kiểu mũ cho phương trình Schrödinger phi tuyến bậc ba trên torus nhiều chiều d $\mathbb{T}^{d}$ T d . Phương pháp này cũng đã được phát triển gần đây trong một bối cảnh rộng hơn của các cây trang trí (Bruned et al. trong Forum Math. Pi 10:1–76, 2022). Phương pháp này là rõ ràng và hiệu quả trong việc triển khai. Tại đây, chúng tôi trình bày một cách phát sinh khác và đưa ra phân tích sai số nghiêm ngặt. Cụ thể, chúng tôi chứng minh sự hội tụ bậc hai trong $H^{\gamma }(\mathbb{T}^{d})$ H γ ( T d ) cho dữ liệu ban đầu trong $H^{\gamma +2}(\mathbb{T}^{d})$ H γ + 2 ( T d ) cho bất kỳ $\gamma > d/2$ γ > d / 2 . Điều này cải thiện công trình trước đó (Knöller et al. trong SIAM J. Numer. Anal. 57:1967–1986, 2019).

Thiết kế của phương pháp dựa trên một phương pháp mới để xấp xỉ tương tác tần số phi tuyến. Điều này cho phép chúng tôi xử lý cấu trúc cộng hưởng phức tạp trong các chiều tùy ý. Các thí nghiệm số phù hợp với kết quả lý thuyết bổ sung cho công trình này.

In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and finding stable and accurate solutions, we employ the Quintic B-spline (QBS) collocation and Tikhonov regularization methods, respectively. The following nonlinear minimization problem is solved using MATLAB. The collected findings demonstrate that accurate and stable solutions can be found.

This paper proposes a local meshless radial basis function (RBF) method to obtain the solution of the two-dimensional time-fractional Sobolev equation. The model is formulated with the Caputo fractional derivative. The method uses the RBF to approximate the spatial operator, and a finite-difference algorithm as the time-stepping approach for the solution in time. The stability of the technique is examined by using the matrix method. Finally, two numerical examples are given to verify the numerical performance and efficiency of the method.

The goal of this paper is to present a new class of operators satisfying the Prešić-type rational η-contraction condition in the setting of usual metric spaces. New fixed point results are also obtained for these operators. Our results generalize, extend, and unify many papers in this direction. Moreover, two examples are derived to support and document our theoretical results. Finally, to strengthen our paper and its contribution to applications, some convergence results for a class of matrix difference equations are investigated.

In this paper, we investigate the generalized fractional system (GFS) with order lying in $(1, 2)$ ( 1 , 2 ) . We present stability analysis of GFS by two methods. First, the stability analysis of that system using the Gronwall–Bellman (G–B) Lemma, the Mittag–Leffler (M–L) function, and the Laplace transform is introduced. Secondly, by the Lyapunov direct method, we study the M–L stability of our system with order lying in $(1, 2)$ ( 1 , 2 ) . Using the modified predictor–corrector method, the solutions of GFSs are calculated and they are more complicated than the classical fractional one. Based on linear feedback control, we investigate a theorem to control the chaotic GFSs with order lying in $(1, 2)$ ( 1 , 2 ) . We present an example to verify the validity of control theorem. We state and prove a theorem to calculate the analytical formula of controllers that are used to achieve synchronization between two different chaotic GFSs. An example to study the synchronization for systems with orders lying in $(1, 2)$ ( 1 , 2 ) is given. We found an agreement between analytical results and numerical simulations.

In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.

Nonlinear fractional difference equations are studied deeply and extensively by many scientists by using fixed-point theorems on different types of function spaces. In this study, we combine fixed-point theory with a set of falling fractional functions in a Banach space to prove the existence and uniqueness of solutions of a class of fractional difference equations. The most important part of this article is devoted to correcting a significant mistake made in the literature in using the power rule by providing further conditions for its validity. Also, we provide specific conditions under which difference equations have attractive solutions and the solutions are also asymptotically stable. Furthermore, we construct some fractional difference examples in order to illustrate the validity of the observed results.