Article
Mathematics, Interdisciplinary Applications
M. Shehata, M. Shokry, R. A. Abd-Elmonem, I. L. El-Kalla
Summary: In this article, the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) is solved using an accelerated form of the Adomian decomposition method (ADM). By converting the NVPFIE to the nonlinear Volterra integral equations based on (alpha,delta, beta)-cut, the transformed system is solved using an accelerated version of the ADM with a new formula for the Adomian polynomial. The sufficient condition for obtaining a unique solution is derived using this new Adomian polynomial, along with error estimates and proof of convergence of the series solution. Numerical cases are presented to demonstrate the effectiveness of this approach.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Mahmoud A. Zaky, Ibrahem G. Ameen, Nermeen A. Elkot, Eid H. Doha
Summary: This paper introduces a spectral collocation method for solving a general class of nonlinear systems of multi-dimensional integral equations. The method uses Legendre spectral quadrature rule to approximate integral terms for high-order accuracy, and establishes the spectral rate of convergence in the L-2-norm, showing exponential decay of error in the approximate solution. Numerical examples validate the theoretical prediction.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Soraya Torkaman, Mohammad Heydari, Ghasem Barid Loghmani
Summary: In this paper, an iterative scheme combining the quasilinearization technique and multi-dimensional linear barycentric rational interpolation is proposed to solve nonlinear multi-dimensional Volterra integral equations. The solution to the nonlinear integral equation is approximated using the quasilinearization method and a collocation method based on multi-dimensional barycentric rational basis functions. The convergence of the iterative sequence and the error estimation of the combined method are investigated, and the efficiency and validity of the proposed method are demonstrated through numerical examples and comparisons with existing methods.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Mahmoud A. Zaky, Ibrahem G. Ameen
Summary: This work presents a spectrally accurate collocation method for solving weakly singular integral equations in high dimensions, utilizing multivariate Jacobi approximation and smoothing transformation to address singularity issues. Rigorous convergence analysis and numerical tests were conducted, confirming the effectiveness of the proposed method for nonsmooth solutions in two dimensions. The results provide a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.
ENGINEERING WITH COMPUTERS
(2021)
Article
Engineering, Multidisciplinary
Mustafa Turkyilmazoglu
Summary: This paper presents a modified Adomian Decomposition Method that improves the convergence control of the traditional ADM by adding a parameterized term with an embedded parameter. By determining the optimal value of the parameter through minimizing the error, the method prevents the failure of the classical ADM and reduces the need to compare results with numerical solutions.
CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES
(2021)
Article
Mathematics, Applied
Fuat Usta
Summary: In this study, the Bernstein approximation method, along with the Riemann-Liouville fractional integral operator, was used to solve both the first and second kind of fractional Volterra integral equations. The proposed technique was shown to be applicable and efficient through convergence analysis and illustrative numerical experiments. All numerical calculations were carried out on a personal computer using MATLAB programs.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaoxia Wen, Jin Huang
Summary: An efficient combination approach grounded on barycentric rational interpolation and Picard iteration is proposed for solving nonlinear stochastic Ito-Volterra integral equations. Theoretical study confirms the error and convergence analysis of the approach, and numerical experiments demonstrate its applicability and efficiency compared with other known numerical methods.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
R. Kaafi, P. Mokhtary, E. Hesameddini
Summary: This paper develops and analyzes a reliable Jacobi Galerkin method for solving cordial Volterra integral equations. The method calculates approximate solutions recursively to avoid solving high-conditioning systems, and it effectively deals with high oscillatory solutions in a long integration domain.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Interdisciplinary Applications
Zi-Qiang Wang, Qin Liu, Jun-Ying Cao
Summary: This paper proposes a higher-order numerical scheme for solving two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. The scheme is based on the modified block-by-block method and involves discretizing the domain into subdomains and using biquadratic Lagrangian interpolation. The convergence of the scheme is rigorously established, and it is proven that the numerical solution converges to the exact solution with the optimal convergence order of O(h(x)(4-alpha) + h(y)(4-beta)) for 0 < alpha,beta < 1. Experimental results with four numerical examples are presented to support the theory and illustrate the efficiency of the proposed method.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Jian Mi, Jin Huang
Summary: This paper presents a method for solving the two-dimensional nonlinear Volterra-Fredholm integral equation using Lagrange interpolation and Legendre-Gauss quadrature formula. The method has the advantage of requiring few collocation points, resulting in small errors, and eliminates the need for integral calculation. It provides proofs for the existence and uniqueness of the original equation under certain conditions, as well as for the solutions of the discrete equations using compact operators theory. Furthermore, the convergence analysis and error estimates are derived, and numerical examples are provided to demonstrate its efficiency and accuracy.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
H. Laeli Dastjerdi, F. Shayanfard
Summary: This paper analyzes the spectral collocation method on a special class of nonlinear Volterra Hammerstein integral equations, discussing convergence analysis and providing examples to validate the theoretical results. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Suleyman Ogrekci, Yasemin Basci, Adil Misir
Summary: This paper investigates the stability of solutions of a general class of nonlinear Volterra integral equations using the Hyers-Ulam and Hyers-Ulam-Rassias stability concepts. The authors improve some well-known results on this problem by applying a fixed point theorem and modifying a widely used technique in similar problems. The paper also provides illustrative examples to showcase the improvement of the mentioned results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Nasibeh Karamollahi, Mohammad Heydari, Ghasem Barid Loghmani
Summary: The study introduces an algorithm based on Hermite interpolation for approximating solutions to second kind Volterra integral equations, applicable to both linear and nonlinear VIEs as well as systems of nonlinear VIEs. The method includes convergence analysis, error estimation, and a multistep form for large intervals. The proposed algorithm provides highly accurate results within acceptable computational times, as demonstrated through various illustrative examples.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics, Applied
Salwa A. Mohamed, Norhan A. Mohamed, Sarah I. Abo-Hashem
Summary: An integration matrix operator is introduced in this work to discretize integro-differential equations, along with a generic differential-integral quadrature method (DIQM). Stability analysis and numerical results demonstrate the exponential convergence and applicability of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Weishan Zheng, Yanping Chen
Summary: In this paper, a spectral method is employed to solve a nonlinear multidimensional Volterra integral equation, and the spectral convergence analysis of the solution is investigated under certain conditions. The numerical example results confirm the theoretical prediction.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Manh Tuan Hoang, Matthias Ehrhardt
Summary: In this paper, a simple approach for solving stiff problems is proposed. Through nonlinear approximation and rigorous mathematical analysis, a class of explicit second-order one-step methods with L-stability and second-order convergence are constructed. The proposed methods generalize and improve existing nonstandard explicit integration schemes, and can be extended to higher-order explicit one-step methods.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Jian Liu, Zengqin Zhao
Summary: In this article, we investigate p(x)-biharmonic equations involving Leray-Lions type operators and Hardy potentials. Some new theorems regarding the existence of generalized solutions are reestablished for such equations when the Leray-Lions type operator and the nonlinearity satisfy suitable hypotheses in variable exponent Lebesgue spaces.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Chengcheng Cheng, Rong Yuan
Summary: This paper investigates the spreading dynamics of a nonlocal diffusion KPP model with free boundaries in time almost periodic media. By applying the novel positive time almost periodic function and satisfying the threshold condition for the kernel function, the unique asymptotic spreading speed of the free boundary problem is accurately expressed.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Xia Wang, Xin Meng, Libin Rong
Summary: In this study, a multiscale model incorporating the modes of infection and types of immune responses of HCV is developed. The basic and immune reproduction numbers are derived and five equilibria are identified. The global asymptotic stability of the equilibria is established using Lyapunov functions, highlighting the significant impact of the reproduction numbers on the overall stability of the model.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Junpu Li, Lan Zhang, Shouyu Cai, Na Li
Summary: This research proposes a regularized singular boundary method for quickly calculating the singularity of the special Green's function at origin. By utilizing the special Green's function and the origin intensity factor technique, an explicit intensity factor suitable for three-dimensional ocean dynamics is derived. The method does not involve singular integrals, resulting in improved computational efficiency and accuracy.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Ying Dong, Shuai Zhang, Yichen Zhang
Summary: This paper investigates a 2D chemotaxis-consumption system with rotation and no-flux-Dirichlet boundary conditions. It proves that under certain conditions on the rotation angle, the corresponding initial-boundary value problem has a classical solution that blows up at a finite time.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Shuhan Yao, Qi Hong, Yuezheng Gong
Summary: In this article, an extended quadratic auxiliary variable method is introduced for a droplet liquid film model. The method shows good numerical solvability and accuracy.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Tong Wang, Binxiang Dai
Summary: This paper investigates the spreading speed and traveling wave of an impulsive reaction-diffusion model with non-monotone birth function and age structure, which models the evolution of annually synchronized emergence of adult population with maturation. The result extends the work recently established in Bai, Lou, and Zhao (J. Nonlinear Sci. 2022). Numerical simulations are conducted to illustrate the findings.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Dinghao Zhu, Xiaodong Zhu
Summary: This paper constructs the soliton solutions of the KdV equation with non-zero background using the Riemann-Hilbert approach. The irregular Riemann-Hilbert problem is first constructed by direct and inverse scattering transform, and then regularized by introducing a novel transformation. The residue theorem is applied to derive the multi-soliton solutions at the simple poles of the Riemann-Hilbert problem. In particular, the interaction dynamics of the two-soliton solution are illustrated by considering their evolutions at different time.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Danhua He, Liguang Xu
Summary: This paper investigates the stability of conformable fractional delay differential systems with impulses. By establishing a conformable fractional Halanay inequality, the paper provides sufficient criteria for the conformable exponential stability of the systems.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Fei Sun, Xiaoli Li, Hongxing Rui
Summary: This paper presents a high-order numerical scheme for solving the compressible wormhole propagation problem. The scheme utilizes the fourth-order implicit Runge-Kutta method and the block-centered finite difference method, along with high-order interpolation technique and cut-off approach to achieve high-order and bound-preserving.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Zhijie Du, Huoyuan Duan
Summary: This study analyzes a direct discretization method for computing the eigenvalues of the Maxwell eigenproblem. It utilizes a specific finite element space and the classical variational formulation, and proves the convergence of the obtained finite element solutions.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Hongliang Li, Pingbing Ming
Summary: This paper proposes an asymptotic-preserving finite element method for solving a fourth order singular perturbation problem, which preserves the asymptotic transition of the underlying partial differential equation. The NZT element is analyzed as a representative, and a linear convergence rate is proved for the solution with sharp boundary layer. Numerical examples in two and three dimensions are consistent with the theoretical prediction.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Shuyang Xue, Yongli Song
Summary: This paper investigates the spatiotemporal dynamics of the memory-based diffusion equation driven by memory delay and nonlocal interaction. The nonlocal interaction, characterized by the given Green function, leads to inhomogeneous steady states with any modes. The joint effect of nonlocal interaction and memory delay can result in spatially inhomogeneous Hopf bifurcation and Turing-Hopf bifurcation.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Baoquan Zhou, Ningzhong Shi
Summary: This paper develops a stochastic SEIS epidemic model perturbed by Black-Karasinski process and investigates the impact of random fluctuations on disease outbreak. The results show that random fluctuations facilitate disease outbreak, and a sufficient condition for disease persistence is established.
APPLIED MATHEMATICS LETTERS
(2024)