Journal of Applied Mathematics and Computing最新文献

In this paper, we establish the existence and uniqueness of solutions for a class of initial value problems involving implicit fractional differential equations with a fractional (Psi )-Caputo derivative on time scales. We employ fixed point theorems by Banach, a nonlinear alternative of Leray-Schauder’s type, and Krasnoselskii’s theorem to establish these results. Finally, we present two examples to demonstrate the effectiveness of the obtained analytical results.

In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique solution that satisfies given initial conditions. Furthermore, we extend our research to investigate the finite-time stability of the system by examining trajectory behavior over a given period. We employ advanced mathematical approaches to systematically prove finite-time stability, providing insights on convergence and stability within the stated interval. Using illustrative examples, we strengthen this all-encompassing examination into the complicated dynamics and stability features of fractionally ordered stochastic systems with time delays. The implications of our results extend to various fields, such as control theory, engineering, and financial mathematics, where understanding the stability of complex systems is crucial.

A numerical scheme based on the Haar wavelets coupled with the nonstandard finite difference scheme is presented to solve the variable-order time-fractional generalized Burgers’ equation (VO-TFGBE). In the proposed technique, firstly, we approximate the time-fractional derivative by the nonstandard finite difference (NSFD) scheme and convert the VO-TFGBE into the nonlinear ordinary differential equation at each time level, and then we apply the Haar wavelet series approximation for the space derivatives. The proposed technique requires only one dimensional Haar wavelet approximation with a significantly smaller number of Haar coefficients to solve time-dependent partial differential equations. The presence of the NSFD scheme provides flexibility to choose different denominator functions and also provides high accuracy for large temporal step sizes. The convergence and stability of the proposed technique are discussed. Some test examples are solved to demonstrate the effectiveness of the technique and validate the theoretical results.

This study describes a new effective technique for solving mixed partial integro-differential equations that are nonlinear with discontinuous kernels (NMPI-DEs). We have used two well-known different numerical techniques, the toeplitz matrix technique (TMT), and the product Nystrom technique (PNT). We have outlined the characteristics of TMT and PNT in both cases, as well as the significance of each approach for characterizing and demystifying the problems’ complexity. These methods have used to convert a system of nonlinear algebraic equations has been derived from the nonlinear Fredholm integral equation (NFIE). Banach’s fixed point theory is employed to investigate the existence and uniqueness of the solution to the nonlinear mixed integral problem. Compared to other approaches, these strategies have shown excellent results in the first instance of being utilized to solve this kind of complex problem. Lastly, a comparison of the two distinct approaches is shown using several cases by using tables and figures. The Maple software has been utilized to compute and obtain all of the numerical results.

The complex q-rung orthopair fuzzy sets are an important way to express uncertain and ambiguous information, and they are superior to the complex fuzzy sets, complex intuitionistic fuzzy sets, complex pythagorean fuzzy sets, and complex fermatean fuzzy sets. This paper extend the notion of q-rung orthopair fuzzy sets to complex q-rung orthopair fuzzy sets. Interaction aggregation operators are often used in various fields to solve multi-attribute decision-making Problems. By utilizing arithmetic and geometric operators, some well-known complex q-rung orthopair fuzzy interaction aggregation operators such as complex q-rung orthopair fuzzy interaction weighted average operator, complex q-rung orthopair fuzzy interaction weighted geometric operator, complex q-rung orthopair fuzzy interaction order weighted operator, complex q-rung orthopair fuzzy interaction order weighted geometric operator, complex q-rung orthopair fuzzy interaction hybrid operator, and complex q-rung orthopair fuzzy interaction hybrid geometric operator have been developed. In addition, some of the unique properties of these newly established operators are investigated. Finally, we explore a decision-making approach to solve multi-attribute decision-making Problem. The viability and flexibility of the suggested technique is explored with the help of a numerical example and the proposed results are compared with several existing approaches.

In this paper, we study preconditioners for all-at-once systems arising from the discretization of time-fractional sub-diffusion equations. Due to the use of high-order accurate formulas in time fractional derivative, the coefficient matrix does not have a Toeplitz structure. We reconstructed the coefficient matrix so that the all-at-once system has a non-symmetric Toeplitz-like structure. Based on the non-symmetric Toplitz-like structure of the new system, we designed a preconditioner that can be quickly diagonalized by discrete sine transform and fast Fourier transform techniques. we show that the spectrum of the preconditioned matrix are clustered around 1. Also, we verified the effectiveness of the proposed preconditioner by numerical experiments.

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

In this paper, to further enhance the efficiency of the improved alternating positive semi-definite splitting (IAPSS) preconditioner proposed by Ren et al. (Numer Algorithms 91:1363–1379, 2022. https://doi.org/10.1007/s11075-022-01305-y), the modified IAPSS preconditioner is established, which can be applied to GMRES method to solve the double saddle point problems. The construction idea of the preconditioner is to modify several sub-matrices in the IAPSS preconditioner. Theoretically, the iteration method generated by the proposed preconditioner is unconditionally convergent for all positive parameters. Furthermore, the selection of the parameters is discussed in detail. Finally, the performance of the preconditioner is verified by the two examples of the liquid crystal director model and the mixed Stokes/Darcy model.

In this paper, we propose an efficient numerical algorithm for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel. In time direction, we use second-order finite difference schemes to discretize the Caputo fractional derivative and also singular integral term. To achieve fully discrete scheme, we apply Galerkin method using generalized Jacobi polynomials as basis, which satisfy essentially all the underlying homogeneous boundary conditions. The proposed method is fast and efficient due to the resulting sparse coefficient matrices. We investigate the error estimate and prove that the method is convergent. Numerical results show the high accuracy and low CPU time of proposed method and confirmed the theoretical ones. Second-order accuracy in time direction and spectral accuracy in space component are also numerically demonstrated by some test problems. Finally we compare the numerical results with the results of other recently methods developed in literature.

For a given graph G, the second Zagreb eccentricity index (xi _2 (G)) is defined as the product of the eccentricities of two adjacent vertex pairs in G. This paper mainly studies the problem of determining the graphs that minimize the second Zagreb eccentricity index among n-vertex bipartite graphs with a fixed number of edges and diameter. To be specific, we determine the sharp lower bound on the second Zagreb eccentricity index over the bipartite graphs of order n in terms of fixed edges and diameter. The extremal graphs attaining these lower bounds are fully characterized.