Article
Mathematics
Ioannis Dassios, Fairouz Tchier, F. M. O. Tawfiq
Summary: This paper discusses a numerical solution method for Abel integral equations based on Muntz-Legendre wavelets, which can effectively solve weakly singular Volterra integral equations. The ability and accuracy of the method are demonstrated through numerical examples.
JOURNAL OF MATHEMATICS
(2022)
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
Multidisciplinary Sciences
Ramses van der Toorn
Summary: Legendre's equation is a key problem in various branches of physics, and its solutions form a linear function space spanned by Legendre functions. However, in physics, only Legendre polynomials are generally accepted as solutions. The quantization of the eigenvalues of Legendre's operator is a consequence of this. In this paper, we present a standalone argument for rejecting all non-polynomial solutions of Legendre's equation in physics and demonstrate that the evenness or oddness of Legendre polynomials is a result of the same premises.
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
Simin Aghaei Amirkhizi, Yaghoub Mahmoudi, Ali Salimi Shamloo
Summary: The paper introduces a new numerical scheme for solving Volterra integral equations of the first kind with piecewise continuous kernels, using operational matrices and shifted Legendre orthogonal polynomials to transform the main equation into a system of linear algebraic equations. Error estimation using bounded operator and numerical examples are provided to demonstrate the accuracy and efficiency of the presented method.
INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
N. M. Temirbekov, L. N. Temirbekova, M. B. Nurmangaliyeva
Summary: This paper reviews new research on approximate methods for solving the first kind Fredholm integral equations, using the Galerkin-Bubnov projection method with Legendre wavelets. Numerical calculations and proven theorems demonstrate the strong sensitivity of the solution to the accuracy of calculating double integrals, and the Galerkin method with Legendre wavelets is shown to be efficient and easy to implement.
TWMS JOURNAL OF PURE AND APPLIED MATHEMATICS
(2022)
Article
Economics
Reza Doostaki, Mohammad Mehdi Hosseini
Summary: This paper presents a numerical solution for evaluating European call and put options using the Black-Scholes partial differential equation. The proposed method is based on finite difference and Legendre wavelets approximation scheme, reducing the Black-Scholes PDE problem to solving a Sylvester equation. The efficiency and capability of the method are demonstrated through numerical results.
COMPUTATIONAL ECONOMICS
(2022)
Article
Mathematics, Applied
S. Behera, S. Saha Ray
Summary: In this article, an effective numerical framework is proposed to solve the pantograph Volterra delay-integro-differential equation. The framework utilizes a new operational matrix scheme based on Muntz-Legendre wavelets. By using the operational matrix of integration and collocation points, the equation is reduced into an explicit system of algebraic equations. The convergence analysis, error estimation, and comparison with other wavelet methods are also conducted.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2023)
Article
Physics, Multidisciplinary
M. R. Elahi, Y. Mahmoudi, A. Salimi Shamloo, M. Jahangiri Rad
Summary: This paper considers the Fredholm integral equations of the first kind with two regular and hypersingular kernels on [-1, 1]. The smoothness of the hypersingular kernel is assumed and no additional conditions are imposed. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to achieve high accurate approximations. The proposed method transforms the integral equation into a system of algebraic equations. Several illustrative examples are provided to demonstrate the efficiency of the method.
Article
Acoustics
Ashish Rayal, Sag R. Verma
Summary: An approximation method with an integral operational matrix based on the Muntz wavelets basis is proposed in this study to solve variational problems, and it is shown to be effective and accurate by examination with illustrative examples.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Automation & Control Systems
Parisa Rahimkhani, Yadollah Ordokhani, Salameh Sedaghat
Summary: In this study, a method based on Muntz-Legendre polynomials (M-LPs) is proposed for solving fractal-fractional 2D optimal control problems. The method involves obtaining operational matrices of fractal-fractional-order derivative, integer-order integration, and derivative of the M-LPs. By applying the M-LPs, operational matrices, and Gauss-Legendre integration, the problem is transformed into a system of algebraic equations which is solved using Newton's iterative method. An error bound is also introduced for the method, and two examples are provided to demonstrate its applicability and validity.
OPTIMAL CONTROL APPLICATIONS & METHODS
(2023)
Article
Mathematics, Applied
Issam Abdennebi, Azedine Rahmoune
Summary: The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation-method without dividing the domain of the integral equation, as usual, into the sub intervals.
NUMERICAL ALGORITHMS
(2023)
Article
Multidisciplinary Sciences
Haifa Bin Jebreen, Carlo Cattani
Summary: This paper presents an effective algorithm using the collocation method and Muntz-Legendre polynomials to solve the fractional gas dynamic equation. By choosing a solution in a finite-dimensional space that satisfies the equation at a set of collocation points, the method shows effectiveness and accuracy in numerical simulations.
Article
Mathematics, Applied
Masoumeh Hosseininia, Mohammad Hossein Heydari, Zakieh Avazzadeh
Summary: This article proposes a hybrid technique for finding approximation solutions of the fractional 2D Sobolev equation. By utilizing Muntz-Legender functions and Muntz-Legender wavelets in the time and spatial directions, the solution of the problem can be approximated. This approach converts the solving process into solving a system of algebraic equations.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Hamid Reza Marzban
Summary: This paper defines and applies a hybrid method of Muntz-Legendre polynomials and block-pulse functions to analyze nonlinear fractional optimal control problems with multiple delays. By using an alternative fractional derivative operator, the primary optimization problem is transformed into an alternative optimization problem involving unknown parameters. Simulation results demonstrate the feasibility and reliability of the proposed method.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Engineering, Multidisciplinary
Mohammad Maleki, Majid Tavassoli Kajani
APPLIED MATHEMATICAL MODELLING
(2015)
Article
Mathematics, Applied
M. Bahmanpour, Majid Tavassoli-Kajani, M. Maleki
COMPUTATIONAL & APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
M. Bahmanpour, Majid Tavassoli-Kajani, M. Maleki
COMPUTATIONAL & APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Mojgan Ghabarpoor, Majid Tavassoli Kajani, Masoud Allame
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2019)
Article
Mathematics, Applied
I. Gholampoor, M. Tavassoli Kajani
NUMERICAL ALGORITHMS
(2020)
Article
Mathematics, Applied
D. Abbaszadeh, M. Tavassoli Kajani, M. Momeni, M. Zahraei, M. Maleki
Summary: This paper presents a Legendre wavelet spectral method for solving a type of fractional Fredholm integro-differential equations, which has applications in computational physics. The method involves a matrix approach for solving linear problems and an iterative matrix method for nonlinear problems, demonstrating efficiency and accuracy in numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
S. Mirshafiee, M. Tavassoli Kajani, H. Sadeghi Goughery
Summary: The hybrid method proposed in this paper combines the advantages of Legendre wavelet spectral collocation scheme and rational approximation using Legendre rational functions, achieving an efficient solution for nonlinear differential equations on semi-infinite domains.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
M. Tavassoli Kajani
Summary: In this paper, two new collocation methods are developed to solve fractional pantograph equations. The first method is a single-domain collocation scheme based on modified Muntz-Legendre polynomials, which provides high accuracy. The second method is a multi-domain collocation scheme equipped with domain decomposition using modified shifted Muntz-Legendre polynomials. The use of shifted Muntz-Legendre polynomials and domain decomposition results in an approximate solution with few collocation points. The accuracy and efficiency of these methods are demonstrated with numerical examples.
MATHEMATICAL SCIENCES
(2023)
Article
Mathematics
M. Ghanbarpoor, M. Tavassoli Kajani
Summary: This study investigates spectral and pseudospectral methods on the half-line using orthogonal systems of shifted Legendre polynomials and rational Legendre functions. A hybrid orthogonal system is introduced, and primary results on hybrid approximations of interpolations and orthogonal projections are established, organizing the theory for solving differential equations on a semi-infinite interval.
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
D. Shirani, M. Tavassoli Kajani, S. Salahshour
Summary: In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. These methods demonstrate high accuracy and efficiency when solving equations involving the derivative of unknown functions in the integral term.
Article
Mathematics
I Gholampoor, M. Tavassoli Kajani
Summary: This paper introduces a novel multi-step pseudo-spectral method for numerically solving an inverse reaction-diffusion equation. The method uses Collocation method and Tikhonov regularization scheme to obtain a numerically stable solution, and the accuracy of the method is demonstrated through numerical examples.
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS
(2021)
Proceedings Paper
Mathematics, Applied
M. Allame, H. Ghasemi, M. Tavassoli Kajani
APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES (AMITANS'15)
(2015)
Proceedings Paper
Mathematics, Applied
M. Dadkhah Tirani, F. Sohrabi, H. Almasieh, M. Tavassoli Kajani
APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES (AMITANS'15)
(2015)
Proceedings Paper
Mathematics, Applied
M. Tavassoli Kajani, I. Gholampoor
APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES (AMITANS'15)
(2015)
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arash Goligerdian, Mahmood Khaksar-e Oshagh
Summary: This paper presents a computational method for simulating more accurate models for population growth with immigration, using integral equations with a delay parameter. The method utilizes Legendre wavelets within the Galerkin scheme as an orthonormal basis and employs the composite Gauss-Legendre quadrature rule for computing integrals. An error bound analysis demonstrates the convergence rate of the method, and various numerical examples are provided to validate the efficiency and accuracy of the technique as well as the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
A. Sreelakshmi, V. P. Shyaman, Ashish Awasthi
Summary: This paper focuses on constructing a lucid and utilitarian approach to solve linear and non-linear two-dimensional partial differential equations. Through testing, it is found that the proposed method is highly applicable and accurate, showing excellent performance in terms of cost-cutting and time efficiency.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Shujiang Tang
Summary: This paper investigates the impact of the structure of local smoothness indicators on the computational performance of the WENO-Z scheme. A new class of two-parameter local smoothness indicators is proposed, which combines the classical WENO-JS and WENO-UD5 schemes and appends the coefficients of higher-order terms. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme achieves optimal accuracy and has higher resolution compared to WENO-JS, WENO-Z, and WENO-UD5.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang
Summary: This paper addresses a class of constrained tensor least squares problems in image restoration and proposes the alternating direction multiplier method (ADMM) to solve them. The convergence analysis of this method is presented. Numerical experiments show the feasibility and effectiveness of the ADMM method for solving constrained tensor least squares problems, and simulation experiments on image restoration are also conducted.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Wanying Mao, Qifeng Zhang, Dinghua Xu, Yinghong Xu
Summary: In this paper, we derive, analyze, and extensively test fourth-order compact difference schemes for the Rosenau equations in one and two dimensions. These schemes are applied under spatial periodic boundary conditions using the double reduction order method and bilinear compact operator. Our results show that these schemes satisfy mass and energy conservation laws and have unique solvability, unconditional convergence, and stability. The convergence order is four in space and two in time under the D infinity-norm. Several numerical examples are provided to support the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hui Yang, Shengfeng Zhu
Summary: In this paper, shape optimization in incompressible Stokes flows is investigated based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed, and numerical algorithms are introduced. An iterative penalty method is used for solving the penalized state and possible adjoint numerically, and it is shown to be more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and error estimates for finite element discretizations of both state and adjoint are provided, and numerical results demonstrate the effectiveness of the optimization algorithms.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arijit Das, Prakrati Kushwah, Jitraj Saha, Mehakpreet Singh
Summary: A new volume and number consistent finite volume scheme is introduced for the numerical solution of a collisional nonlinear breakage problem. The scheme achieves number consistency by introducing a single weight function in the flux formulation. The proposed scheme is efficient and robust, allowing easy coupling with computational fluid dynamics softwares.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
H. Ait el Bhira, M. Kzaz, F. Maach, J. Zerouaoui
Summary: We present an asymptotic method for efficiently computing second-order telegraph equations with high-frequency extrinsic oscillations. The method uses asymptotic expansions in inverse powers of the oscillatory parameter and derives coefficients through either recursion or solving non-oscillatory problems, leading to improved performance as the oscillation frequency increases.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Alexander Zlotnik, Timofey Lomonosov
Summary: This paper studies a three-level explicit in time higher-order vector compact scheme for solving initial-boundary value problems for the n-dimensional wave equation and acoustic wave equation with variable speed of sound. By using additional sought functions to approximate second order non-mixed spatial derivatives of the solution, new stability bounds and error bounds of orders 4 and 3.5 are rigorously proved. Generalizations to nonuniform meshes in space and time are also discussed.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Fengli Yin, Yayun Fu
Summary: This paper develops an explicit energy-preserving scheme for solving the coupled nonlinear Schrodinger equation by combining the Lie-group method and GSAV approaches. The proposed scheme is efficient, accurate, and can preserve the modified energy of the system.
APPLIED NUMERICAL MATHEMATICS
(2024)
