Article
Mathematics, Applied
Lijing Shen, Shuai Zhu, Bingcheng Liu, Zirui Zhang, Yuanda Cui
Summary: In this study, a numerical scheme for solving the nonlinear system of fractional Volterra integro-differential equations using Legendre wavelet is proposed. The Legendre wavelet operational matrix of fractional integration is derived to transform the main system into a system of algebraic equations. The error estimate of the original system is thoroughly investigated, and the proposed method is tested with numerical examples, showing good performance compared to the CAS wavelet method.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Chetna Biswas, Subir Das, Anup Singh, Holm Altenbach
Summary: This article develops a numerical method based on shifted Legendre collocation method using operational matrices to solve a one-dimensional variable-order non-linear partial integro-differential equation. The proposed numerical scheme is validated by comparing the numerical solution with the known exact solution and is illustrated graphically for different variable order derivatives in the presence of different parameter values.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2023)
Article
Mathematics
Haifa Bin Jebreen, Ioannis Dassios
Summary: This paper proposes an efficient algorithm to find an approximate solution to the fractional Fredholm integro-differential equations (FFIDEs) using the wavelet collocation method. By reducing the equation to an integral equation and applying the wavelet collocation method, the algorithm provides accurate and precise results.
Article
Mathematics, Applied
Abhishek Kumar Singh, Mani Mehra
Summary: In this paper, a collocation method is developed using operational matrices based on the Muntz-Legendre polynomial, generated from the fractional basis, to solve the stochastic fractional integro-differential equation (SFIDE). The operational matrices for the involved operators in the SFIDE are introduced to meet the requirements of the numerical method. Additionally, an algorithm using the differential evolution algorithm is proposed to find the Muntz space that provides the best results. Convergence analysis of the proposed method is also studied, and test examples demonstrate the accuracy and efficiency of the numerical method.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Interdisciplinary Applications
Anqi Zhang, Roghayeh Moallem Ganji, Hossein Jafari, Mahluli Naisbitt Ncube, Latifa Agamalieva
Summary: This paper presents a numerical algorithm to solve distributed order integro-differential equations, using matrices derived from shifted Legendre polynomials to approximate the solution and compute numerical values of the polynomial coefficients. Theoretical aspects of error bounds are discussed, and illustrative examples are included to demonstrate the validity and applicability of the method.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Mathematics, Applied
Chuanhua Wu, Ziqiang Wang
Summary: This paper proposes a high-precision numerical algorithm for solving fractional integro-differential equations based on the shifted Legendre polynomials, Gauss-Legendre quadrature rule, and spectral collocation method. The error analysis of this method is provided in detail, and numerical examples demonstrate the exponential convergence of the proposed method.
Article
Mathematics, Interdisciplinary Applications
D. Uma, S. Raja Balachandar, S. G. Venkatesh, K. Balasubramanian, Mantepu Tshepo Masetshaba
Summary: This paper proposes a shifted Legendre polynomial approximations-based stochastic operational matrix of integration method to solve persistent processes-based fractional stochastic differential equations. The operational matrix of integration, stochastic operation matrix, and fractional stochastic operational matrix of the shifted Legendre polynomials are derived. The proposed approach transforms the stochastic differential equation into an algebraic system of (N + 1) equations using the operational matrices. Thorough error analysis in L-2 norm is provided to evaluate the method's accuracy. The applicability, correctness, and accuracy of the proposed method are examined through numerical examples and compared to other methods from the literature, demonstrating its effectiveness and high quality. The error analysis further reveals the method's superiority, achieving a more accurate solution and minimizing errors.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics, Applied
Abhishek Kumar Singh, Mani Mehra, Vaibhav Mehandiratta
Summary: In this work, a new collocation method based on Muntz-Legendre polynomial and operational matrices is proposed to solve variable-order stochastic fractional integro-differential equation. The existence and uniqueness of the solution for the considered problem are proven. The operational matrices are used to convert the equation into a nonlinear system of algebraic equations. The accuracy and efficiency of the proposed method are investigated through numerical experiments, and the advantage of the variable-order model over the constant order model is demonstrated through a real-world example in mathematical finance.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Yu Ping Wang, Emrah Yilmaz, Shahrbanoo Akbarpoor
Summary: This study examines integro-differential equations under separated boundary conditions. It focuses on the inverse nodal problem, using Legendre polynomials and Legendre wavelets method to obtain approximate solutions. The study also presents the error bound of q(x) for continuous derivatives.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
A. B. Deshi, G. A. Gudodagi
Summary: Wavelets are currently important in numerical analysis, especially in solving nonlinear fractional differential equations using the Haar wavelet collocation method. The proposed method provides approximate solutions that are more accurate compared to traditional numerical methods, and increasing the number of collocation points helps reduce errors, as demonstrated in illustrative examples.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Computer Science, Interdisciplinary Applications
S. Behera, S. Saha Ray
Summary: This article proposes an effective approach to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. By introducing two-dimensional wavelets and constructing operational matrices, the equations are transformed into systems of linear and nonlinear algebraic equations. The convergence and error analysis are conducted using the wavelets approximation, and the accuracy and effectiveness of the proposed scheme are verified through illustrative examples.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mathematics, Applied
Sayed Arsalan Sajjadi, Hashem Saberi Najafi, Hossein Aminikhah
Summary: This paper proposes an algorithm to address the non-smooth behavior of solutions of nonlinear fractional Volterra integro-differential equations with weakly singular kernels. The convergence of the algorithm is investigated and four numerical examples are solved to test its efficiency and accuracy.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Physics, Multidisciplinary
Aman Singh, Nikhil Srivastava, Somveer Singh, Vineet Kumar Singh
Summary: This article aims to construct two new schemes for numerically solving the nonlinear weakly singular integro-fractional differential equation in 1D and 2D. These schemes reduce the original equation into a system of nonlinear algebraic equations, greatly simplifying the problem. The proposed schemes are demonstrated on several test examples, confirming their desired accuracy and efficiency.
CHINESE JOURNAL OF PHYSICS
(2022)
Article
Multidisciplinary Sciences
H. Jafari, N. A. Tuan, R. M. Ganji
Summary: In this work, a numerical scheme based on orthogonal basis functions is proposed to solve a general class of pantograph type nonlinear fractional integro-differential equations. The unknown function and its derivatives are expanded in terms of SLPs with unknown coefficients, and several theorems based on SLPs are presented to help achieve the approximate solution of the problem. The main problem is transformed into a system of linear or nonlinear algebraic equations using these theorems and collocation points, and the accuracy and efficiency of the scheme are demonstrated through four illustrative examples.
JOURNAL OF KING SAUD UNIVERSITY SCIENCE
(2021)
Article
Mathematics, Interdisciplinary Applications
Haifa Bin Jebreen
Summary: This paper presents a wavelet collocation method for solving weakly singular integro-differential equations with fractional derivatives. The method reduces the desired equation to a corresponding Volterra integral equation and transforms it into a system of nonlinear algebraic equations using the operational matrix of fractional integration. Numerical simulations demonstrate the efficiency and accuracy of the method.
FRACTAL AND FRACTIONAL
(2023)
Article
Engineering, Multidisciplinary
A. A. Aganin, A. I. Davletshin
Summary: A mathematical model of interaction of weakly non-spherical gas bubbles in liquid is proposed in this paper. The model equations are more accurate and compact compared to existing analogs. Five problems are considered for validation, and the results show good agreement with experimental data and numerical solutions. The model is also used to analyze the behavior of bubbles in different clusters, providing meaningful insights.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Hao Wu, Jie Sun, Wen Peng, Lei Jin, Dianhua Zhang
Summary: This study establishes an analytical model for the coupling of temperature, deformation, and residual stress to explore the mechanism of residual stress formation in hot-rolled strip and how to control it. The accuracy of the model is verified by comparing it with a finite element model, and a method to calculate the critical exit crown ratio to maintain strip flatness is proposed.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Shengwen Tu, Naoki Morita, Tsutomu Fukui, Kazuki Shibanuma
Summary: This study aimed to extend the finite element method to cope with elastic-plastic problems by introducing the s-version FEM. The s-version FEM, which overlays a set of local mesh with fine element size on the conventional FE mesh, simplifies domain discretisation and provides accurate numerical predictions. Previous applications of the s-version FEM were limited to elastic problems, lacking instructions for stress update in plasticity. This study presents detailed instructions and formulations for addressing plasticity problems with the s-version FEM and analyzes a stress concentration problem with linear/nonlinear material properties.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Bo Fan, Zhongmin Wang
Summary: A 3D rotating hyperelastic composite REF model was proposed to analyze the influence of tread structure and rotating angular speed on the vibration characteristics of radial tire. Nonlinear dynamic differential equations and modal equations were established to study the effects of internal pressure, tread pressure sharing ratio, belt structure, and rotating angular speed on the vibration characteristics.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
X. W. Chen, Z. Q. Yue, Wendal Victor Yue
Summary: This paper examines the axisymmetric problem of a flat mixed-mode annular crack near and parallel to an arbitrarily graded interface in functionally graded materials (FGMs). The crack is modeled as plane circular dislocation loop and an efficient solution for dislocation in FGMs is used to calculate the stress field at the crack plane. The analytical solutions of the stress intensity factors are obtained and numerical study is conducted to investigate the fracture mechanics of annular crack in FGMs.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Xumin Guo, Jianfei Gu, Hui Li, Kaihua Sun, Xin Wang, Bingjie Zhang, Rangwei Zhang, Dongwu Gao, Junzhe Lin, Bo Wang, Zhong Luo, Wei Sun, Hui Ma
Summary: In this study, a novel approach combining the transfer matrix method and lumped parameter method is proposed to analyze the vibration response of aero-engine pipelines under base harmonic and random excitations. The characteristics of the pipelines are investigated through simulation and experiments, validating the effectiveness of the proposed method.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Xiangyu Sha, Aizhong Lu, Ning Zhang
Summary: This paper investigates the stress and displacement of a layered soil with a fractional-order viscoelastic model under time-varying loads. The correctness of the solutions is validated using numerical methods and comparison with existing literature. The research findings are of significant importance for exploring soil behavior and its engineering applications under time-varying loads.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Thuy Dong Dang, Thi Kieu My Do, Minh Duc Vu, Ngoc Ly Le, Tho Hung Vu, Hoai Nam Vu
Summary: This paper investigates the nonlinear torsional buckling of corrugated core sandwich toroidal shell segments with functionally graded graphene-reinforced composite (FG-GRC) laminated coatings in temperature change using the Ritz energy method. The results show the significant beneficial effects of FG-GRC laminated coatings and corrugated core on the nonlinear buckling responses of structures.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Zhihao Zhai, Chengbiao Cai, Qinglai Zhang, Shengyang Zhu
Summary: This paper investigates the effect of localized cracks induced by environmental factors on the dynamic performance and service life of ballastless track in high-speed railways. A mathematical approach for forced vibrations of Mindlin plates with a side crack is derived and implemented into a train-track coupled dynamic system. The accuracy of this approach is verified by comparing with simulation and experimental results, and the dynamic behavior of the side crack under different conditions is analyzed.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
James Vidler, Andrei Kotousov, Ching-Tai Ng
Summary: The far-field methodology, developed by J.C. Maxwell, is utilized to estimate the effective third order elastic constants of composite media containing random distribution of spherical particles. The results agree with previous studies and can be applied to homogenization problems in other fields.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Kim Q. Tran, Tien-Dat Hoang, Jaehong Lee, H. Nguyen-Xuan
Summary: This study presents novel frameworks for graphene platelets reinforced functionally graded triply periodic minimal surface (GPLR-FG-TPMS) plates and investigates their performance through static and free vibration analyses. The results show that the mass density framework has potential for comparing different porous cores and provides a low weight and high stiffness-to-weight ratio. Primitive plates exhibit superior performance among thick plates.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Bence Hauck, Andras Szekrenyes
Summary: This study explores several methods for computing the J-integral in laminated composite plate structures with delamination. It introduces two special types of plate finite elements and a numerical algorithm. The study presents compact formulations for calculating the J-integral and applies matrix multiplication to take advantage of plate transition elements. The models and algorithms are applied to case studies and compared with analytical and previously used finite element solutions.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Wu Ce Xing, Jiaxing Wang, Yan Qing Wang
Summary: This paper proposes an effective mathematical model for bolted flange joints to study their vibration characteristics. By modeling the flange and bolted joints, governing equations are derived. Experimental studies confirm that the model can accurately predict the vibration characteristics of multiple-plate structures.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Pingchao Yu, Li Hou, Ke Jiang, Zihan Jiang, Xuanjun Tao
Summary: This paper investigates the imbalance problem in rotating machinery and finds that mass imbalance can induce lateral-torsional coupling vibration. By developing a model and conducting detailed analysis, it is discovered that mass imbalance leads to nonlinear time-varying characteristics and there is no steady-state torsional vibration in small unbalanced rotors. Under largely unbalanced conditions, both resonant and unstable behavior can be observed, and increasing lateral damping can suppress instability and reduce lateral amplitude in the resonance region.
APPLIED MATHEMATICAL MODELLING
(2024)
Article
Engineering, Multidisciplinary
Yong Cao, Ziwen Guo, Yilin Qu
Summary: This paper investigates the mechanically induced electric potential and charge redistribution in a piezoelectric semiconductor cylindrical shell. The results show that doping levels can affect the electric potentials and mechanical displacements, and alter the peak position of the zeroth-order electric potential. The doping level also has an inhibiting effect on the first natural frequency. These findings are crucial for optimizing the design and performance of cylindrical shell-shaped sensors and energy harvesters.
APPLIED MATHEMATICAL MODELLING
(2024)