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Journal articles on the topic 'Polynomial collocation method'

Author: Grafiati

Published: 7 June 2025

Last updated: 9 July 2025

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1

C, Kayelvizhi, and Emimal Kanaga Pushpam A. "Application of Polynomial Collocation Method Based on Successive Integration Technique for Solving Delay Differential Equation in Parkinson's Disease." Indian Journal of Science and Technology 17, no. 2 (2024): 112–19. https://doi.org/10.17485/IJST/v17i2.2573.

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Abstract <strong>Background/Objective:</strong> Parkinson’s disease (PD) is a neurological disorder that is often prevalent in elderly people. This is induced by the reduction or loss of dopamine secretion. The main objective of this work is to apply the polynomial collocation method using successive integration technique for solving delay differential equations (DDEs) arising in PD models. <strong>Methods:</strong> The polynomial collocation method based on successive integration techniques is proposed to obtain approximate solutions of the PD models. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. <strong>Findings:</strong> Numerical examples of two PD models have been considered to demonstrate the efficiency of the proposed method. Numerical simulations of the proposed method are well comparable to the simulation by step method using Picard approximation. <strong>Novelty:</strong> The numerical simulation demonstrates the reliability and efficiency of the proposed polynomial collocation method. The proposed method is very effective, simple, and suitable for solving the nonlinear DDEs model of PD and similar real-world problems exist in different fields of science and engineering. <strong>Keywords:</strong> Polynomial collocation method, Successive Integration Technique, Delay differential equation, Parkinson's disease, Simulation

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2

Molla, Hasib Uddin, and Goutam Saha. "Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 11–25. http://dx.doi.org/10.3329/ganit.v38i0.39782.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

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3

Mirkov, Nikola, Nicola Fabiano, Dušan Nikezić, Vuk Stojiljković, and Milica Ilić. "Symmetries of Bernstein Polynomial Differentiation Matrices and Applications to Initial Value Problems." Symmetry 17, no. 1 (2024): 47. https://doi.org/10.3390/sym17010047.

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In this study, we discuss the symmetries underlying Bernstein polynomial differentiation matrices, as they are used in the collocation method approach to approximate solutions of initial and boundary value problems. The symmetries are brought into connection with those of the Chebyshev pseudospectral method (Chebyshev polynomial differentiation matrices). The treatment discussed here enables a faster and more accurate generation of differentiation matrices. The results are applied in numerical solutions of several initial value problems for the partial differential equation of convection–diffusion reaction type. The method described herein demonstrates the combination of advanced numerical techniques like polynomial interpolation, stability-preserving timestepping, and transformation methods to solve a challenging nonlinear PDE efficiently. The use of Bernstein polynomials offers a high degree of accuracy for spatial discretization, and the CGL nodes improve the stability of the polynomial approximation. This analysis shows that exploiting symmetry in the differentiation matrices, combined with the wise choice of collocation nodes (CGL), leads to both accurate and efficient numerical methods for solving PDEs and accuracy that approach pseudospectral methods that use well-known orthogonal polynomials such as Chebyshev polynomials.

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4

Bin Jebreen, Haifa, and Carlo Cattani. "Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials." Symmetry 15, no. 11 (2023): 2076. http://dx.doi.org/10.3390/sym15112076.

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To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy.

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5

Imani, Ahmad, Azim Aminataei, and Ali Imani. "Collocation Method via Jacobi Polynomials for Solving Nonlinear Ordinary Differential Equations." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/673085.

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We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials.

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6

Pushpam, A. E. K., and C. Kayelvizhi. "Polynomial Collocation Methods Based on Successive Integration Technique for Solving Neutral Delay Differential Equations." Journal of Scientific Research 16, no. 2 (2024): 343–54. http://dx.doi.org/10.3329/jsr.v16i2.65152.

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This paper presents a new approach to using polynomials such as Hermite, Bernoulli, Chebyshev, Fibonacci, and Bessel to solve neutral delay differential equations. The proposed method is based on the truncated polynomial expansion of the function together with collocation points and successive integration techniques. This method reduces the given equation to a system of nonlinear equations with unknown polynomial coefficients which can be easily calculated. The convergence of the proposed method is discussed with several mild conditions. Numerical examples are considered to demonstrate the efficiency of the method. The numerical results reveal that the proposed new approach gives better results than the conventional operational matrix approach of the polynomial collocation method. It demonstrates the reliability and efficiency of this method for solving linear and nonlinear neutral delay differential equations.

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7

Wu, Nan-Jing, and Ting-Kuei Tsay. "A robust local polynomial collocation method." International Journal for Numerical Methods in Engineering 93, no. 4 (2012): 355–75. http://dx.doi.org/10.1002/nme.4380.

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8

C. Kayelvizhi and A. Emimal Kanaga Pushpam. "Subdomain collocation method based on successive integration technique for solving delay differential equations." International Journal of Science and Research Archive 11, no. 2 (2024): 382–90. http://dx.doi.org/10.30574/ijsra.2024.11.2.0429.

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The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.

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9

Leyk, Zbigniew. "A C0-Collocation-like method for elliptic equations on rectangular regions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 38, no. 3 (1997): 368–87. http://dx.doi.org/10.1017/s0334270000000734.

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AbstractWe describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.

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10

Arif, M. Ziaul, Ahmad Kamsyakawuni, and Ikhsanul Halikin. "Modified Chebyshev Collocation Method for Solving Differential Equations." CAUCHY 3, no. 4 (2015): 208. http://dx.doi.org/10.18860/ca.v3i4.2923.

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This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

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11

Hakk, Kristiina, and Arvet Pedas. "NUMERICAL SOLUTIONS AND THEIR SUPERCONVERGENCE FOR WEAKLY SINGULAR INTEGRAL EQUATIONS WITH DISCONTINUOUS COEFFICIENTS." Mathematical Modelling and Analysis 3, no. 1 (1998): 104–13. http://dx.doi.org/10.3846/13926292.1998.9637093.

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The piecewise polynomial collocation method is discussed to solve second kind Fredholm integral equations with weakly singular kernels K (t, s) which may be discontinuous at s = d, d = const. The main result is given in Theorem 4.1. Using special collocation points, error estimates at the collocation points are derived showing a more rapid convergence than the global uniform convergence in the interval of integration available by piecewise polynomials.

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12

Kyato, Denen Dickson, and Terhemen Aboiyar. "EULER POLYNOMIALS COLLOCATION METHOD FOR SOLVING LANE-EMDEN EQUATIONS." FUDMA JOURNAL OF SCIENCES 8, no. 6 (2024): 461–65. https://doi.org/10.33003/fjs-2024-0806-3036.

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In this paper, Euler polynomials were utilized to formulate a collocation method for approximating Lane-Emden equations. The approximation formulations were initialized producing truncation function. The residual functions were constructed by using the truncation function. The collocation points were substituted into the residual function to form system of equations and were solved by Matlab application fsolve or Newton-Raphson methods. The results were tabulated and compared with the Herodt (2004) results for absolute error. It was observed that the Euler polynomial is very accurate and converges faster producing zero error as compared. The study recommended in solving Lane-Emden and higher order equations.

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13

Liang, Hui, and Hermann Brunner. "Collocation methods for integro-differential algebraic equations with index 1." IMA Journal of Numerical Analysis 40, no. 2 (2019): 850–85. http://dx.doi.org/10.1093/imanum/drz010.

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Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

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14

Zhao, Peichen, and Yongling Cheng. "Linear Barycentric Rational Method for Solving Schrodinger Equation." Journal of Mathematics 2021 (October 27, 2021): 1–7. http://dx.doi.org/10.1155/2021/5560700.

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A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.

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15

Rawashdeh, Edris, Dave McDowell, and Leela Rakesh. "The stability of collocation methods for VIDEs of second order." International Journal of Mathematics and Mathematical Sciences 2005, no. 7 (2005): 1049–66. http://dx.doi.org/10.1155/ijmms.2005.1049.

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Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with|λ|=1belong to a1×1Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure.

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16

Journal, Baghdad Science. "An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method." Baghdad Science Journal 15, no. 3 (2018): 344–51. http://dx.doi.org/10.21123/bsj.15.3.344-351.

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The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

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17

Al-Saif, Nahdh S. M., and Ameen Sh Ameen. "Numerical Solution of Mixed Volterra – Fredholm Integral Equation Using the Collocation Method." Baghdad Science Journal 17, no. 3 (2020): 0849. http://dx.doi.org/10.21123/bsj.2020.17.3.0849.

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Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme.

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18

Baleanu, D., A. H. Bhrawy, and T. M. Taha. "A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/413529.

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This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomialsLi(α,β)(x)withx∈Λ=(0,∞),α>−1, andβ>0, andiis the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.

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19

Li, Jin. "Linear barycentric rational collocation method for solving biharmonic equation." Demonstratio Mathematica 55, no. 1 (2022): 587–603. http://dx.doi.org/10.1515/dema-2022-0151.

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Abstract Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

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20

Omole, Ezekiel, A. A. Aigbiremhon, and Abosede Funke Familua. "A THREE-STEP INTERPOLATION TECHNIQUE WITH PERTURBATION TERM FOR DIRECT SOLUTION OF THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS." FUDMA JOURNAL OF SCIENCES 5, no. 2 (2021): 365–76. http://dx.doi.org/10.33003/fjs-2021-0502-556.

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In this paper, we developed a new three-step method for numerical solution of third order ordinary differential equations. Interpolation and collocation methods were used by choosing interpolation points at steps points using power series, while collocation points at step points, using a combination of powers series and perturbation terms gotten from the Legendre polynomials, giving rise to a polynomial of degree and equations. All the analysis on the method derived shows that it is zero-stable, convergent and the region of stability is absolutely stable. Numerical examples were provided to test the performance of the method. Results obtained when compared with existing methods in the literature, shows that the method is accurate and efficient

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21

Rawashdeh, Edris, Dave McDowell, and Leela Rakesh. "The stability of collocation methods for higher-order Volterra integro-differential equations." International Journal of Mathematics and Mathematical Sciences 2005, no. 19 (2005): 3075–89. http://dx.doi.org/10.1155/ijmms.2005.3075.

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The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.

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22

Germider, O. V., and V. N. Popov. "On the Collocation Method in Constructing a Solution to the Volterra Integral Equation of the Second Kind Using Chebyshev and Legendre Polynomials." Bulletin of Irkutsk State University. Series Mathematics 50 (2024): 19–35. https://doi.org/10.26516/1997-7670.2024.50.19.

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The paper proposes a matrix implementation of the collocation method for constructing a solution to Volterra integral equations of the second kind using systems of orthogonal Chebyshev polynomials of the first kind and Legendre polynomials. The integrand in the equations considered in this work is represented as a partial sum of a series for these polynomials. The roots of the Chebyshev and Legendre polynomials are chosen as collocation points. Using matrix and integral transformations, properties of finite sums of products of these polynomials and weight functions at the zeros of the corresponding polynomials with degree equal to the number of nodes, integral equations are reduced to systems of linear algebraic equations for unknown values of the sought functions at these points. As a result, solutions to Volterra integral equations of the second kind are found by polynomial interpolations of the obtained function values at collocation points using inverse matrices, the elements of which are written on the basis of orthogonal relations for these polynomials. In the presented work, the elements of integral matrices are also given in explicit form. Error estimates for the constructed solutions with respect to the infinite norm are obtained. The results of computational experiments are presented, which demonstrate the effectiveness of the collocation method used.

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23

Adamu, Samuel, O. O. Aduroja, and K. Bitrus. "NUMERICAL SOLUTION TO OPTIMAL CONTROL PROBLEMS USING COLLOCATION METHOD VIA PONTRYAGIN’S PRINCIPLE." FUDMA JOURNAL OF SCIENCES 7, no. 5 (2023): 228–33. http://dx.doi.org/10.33003/fjs-2023-0705-2016.

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In this study, Lucas polynomial approximate solution is considered to develop a collocation technique for solving optimal control problems with implementation in block using forward backward sweep method. The collocation block method developed is stable and convergent. The method is implemented using MATLAB code, and the examples show that forward-backward sweep methods with the collocation method is an efficient technique for solving optimal control problems as compared with some existing methods.

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Gu, Zhendong, and Yanping Chen. "Piecewise Legendre spectral-collocation method for Volterra integro-differential equations." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 231–49. http://dx.doi.org/10.1112/s1461157014000485.

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Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in$h^{m}N^{-m}$-version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.

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Abdelkawy, M. A. "A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (2018): 781–92. http://dx.doi.org/10.1515/ijnsns-2018-0111.

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AbstractIn this work, shifted fractional-order Jacobi orthogonal function in the interval $[0,\mathcal{T}]$ is outputted of the classical Jacobi polynomial (see Definition 2.3). Also, we list and derive some facts related to the shifted fractional-order Jacobi orthogonal function. Spectral collocation techniques are addressed to solve the multidimensional distributed-order diffusion equations (MDODEs). A mixed of shifted Jacobi polynomials and shifted fractional order Jacobi orthogonal functions are used as basis functions to adapt the spatial and temporal discretizations, respectively. Based on the selected basis, a spectral collocation method is listed to approximate the MDODEs. By means of the selected basis functions, the given conditions are automatically satisfied. We conclude with the application of spectral collocation method for multi-dimensional distributed-order diffusion equations.

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Nirmala, A. N., and S. Kumbinarasaiah. "An intriguing numerical strategy for Zakharov–Kuznetsov equation through graph-theoretic polynomials." Physica Scripta 99, no. 9 (2024): 095267. http://dx.doi.org/10.1088/1402-4896/ad6c8e.

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Abstract This paper explores graph-theoretic polynomials to find the approximate solution of the (2+1)D Time-fractional Zakharov-Kuznetsov(TF-Z-K) equation. The Zakharov-Kuznetsov equations govern the behavior of nonlinear acoustic waves in the plasma of hot isothermal electrons and cold ions in the presence of a homogeneous magnetic field. Independence polynomials of the Ladder-Rung graph serve as the polynomial approximation for the suggested Independence Polynomial Collocation Method (IPCM). The Caputo fractional derivatives are adopted to determine the fractional derivatives in the TF-Z-K equation. The TF-Z-K equation is converted into a system of nonlinear algebraic equations using the collocation points in IPCM. The Newton-Raphson approach yields the solution of the suggested method by solving the resulting system. We’ve compared a few scenarios with the tangible outcomes to validate the procedure. Quantitative outcomes match the current findings and validate the exactness of IPCM compared t o the recent numerical and semi-analytical approaches in the literature.

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27

Boubaker, K. "A Confirmed Model to Polymer Core-Shell Structured Nanofibers Deposited via Coaxial Electrospinning." ISRN Polymer Science 2012 (October 14, 2012): 1–6. http://dx.doi.org/10.5402/2012/603108.

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A model to core-shell structured polymer nanofibers deposited via coaxial electrospinning is presented. Investigations are based on a modified Jacobi-Gauss collocation spectral method, proposed along with the Boubaker Polynomials Expansion Scheme (BPES), for providing solution to a nonlinear Lane-Emden-type equation. The spatial approximation has been based on shifted Jacobi polynomials with was n the polynomial degree. The Boubaker Polynomials Expansion Scheme (BPES) main features, concerning the embedded boundary conditions, have been outlined. The modified Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique, and a comparison is made with existing results. It has been revealed that both methods are easy to implement and yield very accurate results.

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Germider, Oksana V., and Vasilii N. Popov. "About the integral approach using the collocation method." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 88 (2024): 14–25. http://dx.doi.org/10.17223/19988621/88/2.

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The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

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Ku, Cheng-Yu, Jing-En Xiao, and Chih-Yu Liu. "A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations." Mathematics 8, no. 2 (2020): 270. http://dx.doi.org/10.3390/math8020270.

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In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

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Shalangwa, Albert A., Matthew R. Odekunle, and Solomon O. Adee. "Numerical Solution of 2D Nonlinear Volterra-Fredholm Integral Equations using Polynomial Collocation Method." International Journal of Development Mathematics (IJDM) 2, no. 1 (2025): 022–31. https://doi.org/10.62054/ijdm/0201.02.

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In this research, polynomial collocation method was used to develop and implement numerical solutions of nonlinear two-dimensional (2D) mixed Volterra-Fredholm integral equations. The Integral equation was transform into systems of algebraic equations using standard collocation points with Bernstein polynomial as a basis function and then solves the nonlinear algebraic equations using Newton-Rhapson method. The analysis of the developed method was investigated and the solution was found to be unique and convergent. To illustrate the efficiency, simplicity, and accuracy of the approach, illustrative examples are provided which shows that the method outperforms the other methods

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Kumar, Sunil. "The numerical solution of Hammerstein equations by a method based on polynomial collocation." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 31, no. 3 (1990): 319–29. http://dx.doi.org/10.1017/s0334270000006676.

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AbstractIn recent papers we have considered the numerical solution of the Hammerstein equationby a method which first applies the standard collocation procedure to an equivalent equation for z(t):= g(t, y(t)), and then obtains an approximation to y by use of the equationIn this paper we approximate z by a polynomial zn of degree ≤ n − 1, with coefficients determined by collocation at the zeros of the nth degree Chebyshev polynomial of the first kind. We then define the approximation to y to beand establish that, under suitable conditions, , uniformly in t.

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Wang, Houjun, John P. Boyd, and Rashid A. Akmaev. "On computation of Hough functions." Geoscientific Model Development 9, no. 4 (2016): 1477–88. http://dx.doi.org/10.5194/gmd-9-1477-2016.

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Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

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Khudhair Abbas, S., S. Sohrabi, and H. Ranjbar. "Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method." Journal of Mathematics 2023 (November 6, 2023): 1–12. http://dx.doi.org/10.1155/2023/9220664.

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This paper deals with the numerical solution of a class of highly oscillatory Volterra integral equations by collocation methods based on the exponential fitting technique. By reviewing the oscillatory structures of solutions of these problems, we construct an exponential fitting collocation method which is best tuned to capture the qualitative behaviour of the solution of these equations. We also investigate the convergence properties of the proposed collocation solution based on the interpolation remainder. Some numerical examples are provided which illustrate the efficiency and accuracy of the proposed method and confirm its superiority over the polynomial collocation methods.

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Sloan, I. H., and E. P. Stephan. "Collocation with Chebyshev polynomials for Symm's integral equation on an interval." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 34, no. 2 (1992): 199–211. http://dx.doi.org/10.1017/s0334270000008729.

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AbstractA collocation method for Symm's integral equation on an interval (a first-kind integral equation with logarithmic kernel), in which the basis functions are Chebyshev polynomials multiplied by an appropriate singular function and the collocation points are Chebyshev points, is analysed. The novel feature lies in the analysis, which introduces Sobolev norms that respect the singularity structure of the exact solution at the ends of the interval. The rate of convergence is shown to be faster than any negative power of n, the degree of the polynomial space, if the driving term is smooth.

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Bataineh, Ahmad Sami, Osman Rasit Isik, Moa’ath Oqielat, and Ishak Hashim. "An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs." Mathematics 9, no. 4 (2021): 425. http://dx.doi.org/10.3390/math9040425.

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In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

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Luga, Tersoo, Sunday Simon Isah, and Vershima Benjamin Iyorter. "Laguerre collocation method for solving higher order linear boundary value problems." Engineering and Applied Science Letters 4, no. 1 (2021): 42–49. https://doi.org/10.30538/psrp-easl2021.0060.

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Collocation methods are efficient approximate methods developed by utilizing suitable set of functions known as trial or basis functions. These methods are used for solving differential equations, integral equations and integro-differential equations, etc. In this study, the Laguerre polynomial of degree 10 is used as a basis function to propose a collocation method for solving higher order linear ordinary differential equations. Four examples on , , and order ordinary differential equations are selected to illustrate the effectiveness of the method. The numerical results show that the proposed collocation method is easy and straightforward to implement, nevertheless, it is very accurate.

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Li, Jin. "Barycentric rational collocation method for semi-infinite domain problems." AIMS Mathematics 8, no. 4 (2023): 8756–71. http://dx.doi.org/10.3934/math.2023439.

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<abstract><p>The barycentric rational collocation method for solving semi-infinite domain problems is presented. Following the barycentric interpolation method of rational polynomial and Chebyshev polynomial, matrix equation is obtained from discrete semi-infinite domain problem. Truncation method and transformation method are presented to solve linear and nonlinear differential equation defined on the semi-infinite domain problems. At last, three numerical examples are presented to valid our theoretical analysis.</p></abstract>

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38

Pedas, Arvet. "POLYNOMIAL SPLINE COLLOCATION METHOD FOR NONLINEAR TWO‐DIMENSIONAL WEAKLY SINGULAR INTEGRAL EQUATIONS." Mathematical Modelling and Analysis 2, no. 1 (1997): 122–29. http://dx.doi.org/10.3846/13926292.1997.9637075.

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Yap, Lee Ken, Fudziah Ismail, and Norazak Senu. "An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/549597.

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The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.

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Guo, Ling, Yongle Liu та Liang Yan. "Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions". Numerical Mathematics: Theory, Methods and Applications 10, № 4 (2017): 775–97. http://dx.doi.org/10.4208/nmtma.2017.0001.

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AbstractIn this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

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Yap, Lee Ken, and Fudziah Ismail. "Block Hybrid Collocation Method with Application to Fourth Order Differential Equations." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/561489.

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The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.

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42

Wu, Qinghua. "The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/172327.

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A method for approximating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatory trigonometric kernel is presented. The unknown function is approximated by expansion of Chebychev polynomial and the coefficients are determinated by classical collocation method. Due to the highly oscillatory kernels of integral equation, the discretised collocation equation will give rise to the computation of oscillatory integrals. These integrals are calculated by using recursion formula derived from the fundamental recurrence relation of Chebyshev polynomial. The effectiveness and accuracy of the proposed method are tested by numerical examples.

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BECK, JOAKIM, RAUL TEMPONE, FABIO NOBILE, and LORENZO TAMELLINI. "ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS." Mathematical Models and Methods in Applied Sciences 22, no. 09 (2012): 1250023. http://dx.doi.org/10.1142/s0218202512500236.

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In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

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44

Parts, I., and A. Pedas. "COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 8, no. 4 (2003): 315–28. http://dx.doi.org/10.3846/13926292.2003.9637233.

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A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

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45

SAIRA, Wen-Xiu Ma, and Guidong Liu. "A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations." Fractal and Fractional 8, no. 2 (2024): 80. http://dx.doi.org/10.3390/fractalfract8020080.

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The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

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Oja, P., and M. Tarang. "STABILITY OF PIECEWISE POLYNOMIAL COLLOCATION FOR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 6, no. 2 (2001): 310–20. http://dx.doi.org/10.3846/13926292.2001.9637170.

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Numerical stability of the spline collocation method by piecewise polynomials for Volterra integro‐differential equations is investigated. Stability conditions depending on collocation parameters and also on parameters of certain test equation are obtained. Results of several numerical tests are presented supporting theoretical results.

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Mirkov, Nikola, and Boško Rašuo. "Bernstein Polynomial Collocation Method for Elliptic Boundary Value Problems." PAMM 13, no. 1 (2013): 421–22. http://dx.doi.org/10.1002/pamm.201310206.

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48

Loh, Jian Rong, Chang Phang, and Abdulnasir Isah. "New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations." Advances in Mathematical Physics 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/3821870.

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It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.

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Cao, Wen, and Yufeng Xu. "Collocation Method for Optimal Control of a Fractional Distributed System." Fractal and Fractional 6, no. 10 (2022): 594. http://dx.doi.org/10.3390/fractalfract6100594.

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In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases.

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Ganaie, Ishfaq Ahmad, Shelly Arora, and V. K. Kukreja. "Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions." International Journal of Engineering Mathematics 2014 (February 24, 2014): 1–8. http://dx.doi.org/10.1155/2014/365209.

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Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.

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