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Convergence Analysis For Numerical Solution of Integral Equations Using Galarkin Method with two Orthogonal Polynomials

Authors

Ezatullah Haleem, Ahad Khan Pyawarai, MOHAMMAD DAUD AHMADZAI, IRSHAD SALARZAI
Type: Article
Publication Date: 2025-04-30
Citations: 0
DOI: https://doi.org/10.47760/cognizance.2025.v05i04.032

Abstract

This research explores the role of orthogonal, Chebyshev, and Hermite polynomials in the numerical solution of integral equations, focusing on convergence analysis. The study employs the Galerkin method to solve Volterra integral equations, providing a numerical approach that ensures accuracy and efficiency. The research emphasizes the importance of integral equations in various scientific and engineering applications, highlighting their transformation from differential equations in physics, biology, and chemistry. MATLAB was used to implement the proposed numerical methods, verifying convergence through examples and erro r estimation. Results demonstrate that the Galerkin method, combined with orthogonal polynomials, effectively approximates solutions with high accuracy. This study contributes to the field of numerical analysis by offering a reliable technique for solving integral equations with reduced computational complexity.

Topics

Locations

  • Cognizance Journal of Multidisciplinary Studies
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Numerical Solutions of Volterra Integral Equations Through Galerkin Weighted Residual Method with Charlier Polynomials

2024-12-30
Md. Shafiqul Islam , Kazi Mohammad Nazib , Md. A. Rahman
Volterra Integral Equations (VIEs) are a significant class of integral equations with broad applications in various fields, such as mathematical physics, engineering, biology, economics, and more. In this paper, we ā€¦ Volterra Integral Equations (VIEs) are a significant class of integral equations with broad applications in various fields, such as mathematical physics, engineering, biology, economics, and more. In this paper, we numerically solve the linear VIEs of both the first and second kind, with both regular and singular kernels, using the Galerkin Weighted Residual Method. Actually, we derive a straightforward and efficient matrix formulation by the Galerkin Method for each type of VIE, employing piecewise Charlier polynomials as the basis functions in the trial solution. Several numerical examples are tested to verify the effectiveness of the proposed method. The numerical results obtained by the proposed method converge monotonically to the exact solutions and, in some cases, achieve the exact solution. In addition, the proposed Charlier polynomials-based Galerkin method significantly outperforms other state-of-the-art methods for the numerical solution of VIEs.

Application of New Orthogonal Shifted Vieta-Lucas Polynomials in Solving Integro-Differential Equations: Methods and Results

2025-02-04
Mohsen Riahi Beni
This article presents an innovative method for solving linear and nonlinear integro-differential equations using Vieta-Lucas polynomials as basis functions, combined with the Galerkin method. Initially, these polynomials are transformed within ā€¦ This article presents an innovative method for solving linear and nonlinear integro-differential equations using Vieta-Lucas polynomials as basis functions, combined with the Galerkin method. Initially, these polynomials are transformed within an arbitrary interval, and their orthogonality is utilized to approximate each function in the equation. A key aspect of this study is the detailed expression of the weight function and orthogonality conditions of these polynomials across any interval. Leveraging these properties and the Galerkin method, the integro-differential equation is converted into a system of algebraic equations. Error estimation is thoroughly investigated through several lemmas and theorems, and the existence and uniqueness of the solution are proven. Finally, numerical tests are conducted using Maple software to validate the accuracy and effectiveness of the proposed method, with comparative analyses demonstrating its superiority over existing techniques.

Using Hermite Polynomials to Solve Volterra-Fredholm Integral Equation of the Second Kind

2024-08-28
Salima Kehali , Amina Khirani
The objective of this study is to solve Linear Volterra-Fredholm Integral Equations of the second kind numerically using Hermite polynomials.We will present an approximate solution as a series that converges ā€¦ The objective of this study is to solve Linear Volterra-Fredholm Integral Equations of the second kind numerically using Hermite polynomials.We will present an approximate solution as a series that converges towards the exact solution.Several examples are provided to illustrate the numerical results, specifically comparing the exact and numerical solutions.These comparisons are shown in tables, demonstrating that the error between the exact and numerical solutions is negligible.Additionally, diagrams highlight how closely the numerical solution matches the exact solution, underscoring the accuracy of the grouping method used to solve the Volterra-Fredholm Integral Equation with the MATLAB program.This method is noted for its simplicity, speed, and high accuracy in obtaining numerical results.
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Hermite Polynomial and Least-Squares Technique for Solving Integro-differential Equations

2022-03-21
Ahsan Fayez Shoushan
The goal of this project is to offer a new technique for solving integro-differential equations (IDEs) with mixed circumstances, which is based on the Hermite polynomial and the Least-Squares Technique ā€¦ The goal of this project is to offer a new technique for solving integro-differential equations (IDEs) with mixed circumstances, which is based on the Hermite polynomial and the Least-Squares Technique (LST). Three examples will be given to demonstrate how the suggested technique works. The numerical results were utilized to demonstrate the correctness and efficiency of the existing method, and all calculations were carried out with the help of the MATLAB R2018b program.
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A Refined Spectral Galerkin Approach Leveraging Romanovskiā€“Jacobi Polynomials for Differential Equations

2025-04-29
Ramy M. Hafez , M. A. Abdelkawy , H. M. Ahmed
This study explores the application of Romanovskiā€“Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions ā€¦ This study explores the application of Romanovskiā€“Jacobi polynomials (RJPs) in spectral Galerkin methods (SGMs) for solving differential equations (DEs). It uses a suitable class of modified RJPs as basis functions that meet the homogeneous initial conditions (ICs) given. We derive spectral Galerkin schemes based on modified RJP expansions to solve three models of high-order ordinary differential equations (ODEs) and partial differential equations (PDEs) of first and second orders with ICs. We provide theoretical assurances of the treatmentā€™s efficacy by validating its convergent and error investigations. The method achieves enhanced accuracy, spectral convergence, and computational efficiency. Numerical experiments demonstrate the robustness of this approach in addressing complex physical and engineering problems, highlighting its potential as a powerful tool to obtain accurate numerical solutions for various types of DEs. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors.

Fractional Variational Orthogonal Collocation Method for the Solution of Fractional Fredholm Integro-Differential Equation Using Mamadu-Njoseh Polynomials

2022-03-01
Jonathan Tsetimi , Ebimene James Mamadu
The use of orthogonal polynomials as basis functions via a suitable approximation scheme for the solution of many problems in science and technology has been on the increase and quite ā€¦ The use of orthogonal polynomials as basis functions via a suitable approximation scheme for the solution of many problems in science and technology has been on the increase and quite fascinating. In many numerical schemes, the convergence depends solely on the nature of the basis function adopted. The Mamadu-Njoseh polynomials are orthogonal polynomials developed in 2016 with reference to the weight function, <img src=image/13426333_01.gif> which bears the same convergence rate as that of Chebyshev polynomials. Thus, in this paper, the fractional variational orthogonal collocation method (FVOCM) is proposed for the solution of fractional Fredholm integro-differential equation using Mamadu-Njoseh polynomials (MNP) as basis functions. Here, the proposed method is an elegant mixture of the variational iteration method (VIM) and the orthogonal collocation method (OCM). The VIM is one of the popular methods available to researchers in seeking the solution to both linear and nonlinear differential problems requiring neither linearization nor perturbation to arrive at the required solution. Collocating at the roots of orthogonal polynomials gives birth to the OCM. For the proposed method, the VIM is initiated to generate the required approximations whereby producing the series <img src=image/13426333_02.gif> which is collocated orthogonally to derive the unknown parameters. The numerical results show that the method derives a high accurate and reliable approximation with a high convergence rate. We have also presented the existence and uniqueness of solution of the method. All computational frameworks in this research are performed via MAPLE 18 software.
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Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

2021-10-05
Asma Akter Akhia , Goutam Saha
In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of ā€¦ In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1ā€“14
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Solution of volterra-fredholm integro-differential equations using Chebyshev least square method

2025-05-12
A.F. Adebis , Kamilu Adedokun Okunola
This paper investigates the application of the least squares method for obtaining numerical solutions to Volterra-Fredholm integrodifferential equations. The least squares method is a wellestablished approach for solving integral equations, ā€¦ This paper investigates the application of the least squares method for obtaining numerical solutions to Volterra-Fredholm integrodifferential equations. The least squares method is a wellestablished approach for solving integral equations, and in this study, it is utilized to find approximate solutions to such equations. To enhance the accuracy of the solutions, Chebyshev polynomials are used as basis functions for the approximation process. These polynomials are chosen due to their favorable convergence properties and their ability to provide accurate approximations over a wide range of problems. Several examples are included in this study to demonstrate the effectiveness and reliability of the proposed method. The numerical results obtained using the least squares method with Chebyshev polynomial approximations are compared with exact solutions, showing excellent agreement. The outcomes of this study indicate that the method is both efficient and reliable for solving Volterra-Fredholm integro-differential equations, offering a robust approach for practical applications.

Precision and efficiency of an interpolation approach to weakly singular integral equations

2024-01-11
Imtiyaz Ahmad Bhat , Lakshmi Narayan Mishra , Vishnu Narayan Mishra +2 more
Purpose This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and ā€¦ Purpose This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm. Design/methodology/approach The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations. Findings Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approachā€™s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software. Research limitations/implications The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement. Practical implications There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively. Social implications This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution. Originality/value To the best of the authorsā€™ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

Numerical treatment of linear Volterra integro differential equations using variational iteration algorithm with collocation

2024-04-30
Ikechukwu Jackson Otaide , Matthew Olanrewaju Oluwayemi
To numerically solve the linear Volterra integro-differential equation, this study employs fourth-kind Chebyshev polynomials and the variational iteration algorithm with collocation, which is a combination of the variational iteration strategy ā€¦ To numerically solve the linear Volterra integro-differential equation, this study employs fourth-kind Chebyshev polynomials and the variational iteration algorithm with collocation, which is a combination of the variational iteration strategy and collocation technique. By applying fourth-kind Chebyshev polynomials to the variational iteration method with collocation for solving Volterra integro-differential equations, mathematical problems with a broad range of multidisciplinary applications are addressed, and numerical techniques that produce more accurate and efficient results are developed. The recommended method is then used, and the fourth-kind Chebyshev polynomials generated for the given integro-differential equation serve as the trial functions for the approximation. As a result, the suggested method's significance probably goes beyond a particular equation or application, as it contributes to the larger field of mathematical modeling and numerical analysis. Research methods employing a variational iteration algorithm with collocation aim to provide general techniques that can be applied to a wide range of problems. Additionally, numerical examples were provided to highlight the applicability and dependability of the proposed methodology. The mathematical computations were carried out using the Maple 18 software.

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

2023-01-01
Omotayo Adebayo Taiwo , Liman Kibokun Alhassan , Olutunde Samuel Odetunde +1 more
This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions ā€¦ This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.
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A Combination of the Orthogonal Polynomials with Least ā€“Squares Method for Solving High-Orders Fredholm-Volterra Integro-Differential Equations

2021-01-07
Hameeda Oda Al-Humedi , Ahsan Fayez Shoushan
This study introduced new technique which is based on a combination of the least-squares technique (LST) with Chebyshev and Legendre polynomials for finding the approximate solutions of higher-order linear Fredholm-Volterra ā€¦ This study introduced new technique which is based on a combination of the least-squares technique (LST) with Chebyshev and Legendre polynomials for finding the approximate solutions of higher-order linear Fredholm-Volterra integro-differential equations (FVIDEs) subject to the mixed conditions. Two examples of second and third-order linear FVIDEs are considered to illustrate the proposed method, the numerical results are comprised to demonstrate the validity and applicability of this technique, and comparisons with the exact solution are made. These results have shown that the competence and accuracy of the present technique.
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TOUCHARD POLYNOMIALS APPROACH FOR SOLVING NON- LINEAR FREDHOLM INTEGRO- DIFFERENTIAL EQUATION OF SECOND TYPE

2024-03-01
Jalil Talab Abdullah , Hayder Majeed ABBAS , Samah Mohammed ALI +1 more
A numerical approach to solve Non- linear Fredholm integro- differential (NLFID) equation of the 1st order and 2nd type has been proposed. The approach was based upon Touchard polynomials. The ā€¦ A numerical approach to solve Non- linear Fredholm integro- differential (NLFID) equation of the 1st order and 2nd type has been proposed. The approach was based upon Touchard polynomials. The non-linear Fredholm integro- differential equation was changed into a system of non- linear algebraic equations and solved using Newton repeating approach. The proposed approach was evaluated by displaying three numerical problems, and the approximate numerical solutions were compared with exact solution and four methods in the literature. MATLAB R2018b was used to perform all calculations and graphs.

Computational methods for solving nonlinear ordinary differential equations arising in engineering and applied sciences

2023-08-30
Othman Mahdi Salih , M. A. AL-Jawary
In this paper, the computational method (CM) based on the standard polynomials has been implemented to solve some nonlinear differential equations arising in engineering and applied sciences. Moreover, novel computational ā€¦ In this paper, the computational method (CM) based on the standard polynomials has been implemented to solve some nonlinear differential equations arising in engineering and applied sciences. Moreover, novel computational methods have been developed in this study by orthogonal base functions, namely Hermite, Legendre, and Bernstein polynomials. The nonlinear problem is successfully converted into a nonlinear algebraic system of equations, which are then solved by MathematicaĀ®12. The developed computational methods (D-CMs) have been applied to solve three applications involving well-known nonlinear problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the methods has been presented. In addition, the maximum error remainder () has been computed to demonstrate the accuracy of the proposed methods. The results persuasively prove that CM and D-CMs are reliable and accurate in obtaining the approximate solutions to the problems, with obvious superiority in accuracy for D-CMs than for CM.
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A Novel Algorithm for Solving Fredholm Integro-Differential Equations Using Lucas Polynomials

2025-02-11
Soomiyol Mrumun Comfort , Kamoh Nathaniel Mahwash , Joshua Sunday
Fredholm integro-differential equations play a crucial role in mathematical modelling across various disciplines, including physics, biology, and finance. In this paper, Fredholm integro-differential equations are solved using the derivative of ā€¦ Fredholm integro-differential equations play a crucial role in mathematical modelling across various disciplines, including physics, biology, and finance. In this paper, Fredholm integro-differential equations are solved using the derivative of the Lucas polynomials in matrix form. The equation is first transformed into systems of nonlinear algebraic equations using the Lucas polynomials. The unknown parameters required for approximating the solution of Fredholm integro-differential equations are then determined using Gaussian elimination. The method has proven to be an active and dependable technique for solving the Fredholm integro-differential equation of any order by updating the matrix of Lucas polynomials. Additionally, the technique is successfully applied to a mixed Fredholm-Volterra integro differential equation demonstrating its versatility. Comparative analysis with some existing methods highlights the improved accuracy and efficiency of the proposed approach. Numerical experiments, including benchmark problems from the literature, confirm the validity and applicability of the technique, achieving lower error margins than conventional methods.
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Application of the Galerkin Method with Chebyshev Polynomials for Solving the Integral Equation

2017-09-30
Mohamed Elarbi Benattia , Kacem Belghaba

Least Squares Approximation Method for Estimation of Volterra Fractional Integro-differential equation using Hermite Polynomial as the basis functions

2024-10-24
Ayodeji Quadri Akinsanya , Gabriel Agholor
This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate ā€¦ This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate solution of degree N, which is then substituted into the fractional-order integro-differential equation under investigation. After evaluating the integral, the equation is rearranged to isolate one side, allowing the application of the least squares method. Three examples were solved using this approach. In Example 1, the numerical results for Ī±=0.9 and Ī±=0.8 were compared to the exact solution for Ī±=1. In Examples 2 and 3, the results for Ī±=1.9 and Ī±=1.8 were compared to the exact solution for Ī±=2. These comparisons showed favorable alignment with the exact solutions. The numerical results and graphical illustrations demonstrate the validity, competence, and accuracy of the proposed method.
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Perturbed Galerkin Method for Solving Integro-Differential Equations

2022-04-15
Kazeem Issa , Jafar Biazar , T. O. Agboola +1 more
In this paper, perturbed Galerkin method is proposed to find numerical solution of an integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions which transform the integro-differential equation ā€¦ In this paper, perturbed Galerkin method is proposed to find numerical solution of an integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions which transform the integro-differential equation into a system of linear equations. The systems of linear equations are then solved to obtain the approximate solution. Examples to justify the effectiveness and accuracy of the method are presented and their numerical results are compared with Galerkinā€™s method, Taylorā€™s series method, and Tauā€™s method which provide validation for the proposed approach. The errors obtained justify the effectiveness and accuracy of the method.
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Numerical solution of ordinary differential equations using Newton-Raphson method with Numpy and Autograd: Accuracy, convergence, and performance analysis

2025-05-30
Choon Kit Chan , Pankaj Dumka , Chandrakant Sonawane +2 more
The numerical solution of ordinary differential equations (ODEs) using the Newton-Raphson approach is investigated in this work. The aim is to evaluate, in solving first- and second-order ODEs, the accuracy, ā€¦ The numerical solution of ordinary differential equations (ODEs) using the Newton-Raphson approach is investigated in this work. The aim is to evaluate, in solving first- and second-order ODEs, the accuracy, convergence, and limits of this approach. This Python-based approach uses Autograd for automatic differentiation and NumPy for effective array operations. Several case studies are analyzed, including various ODE challenges. The efficiency of the approach is assessed by comparing numerical findings with analytical solutions. In many situations, the Newton-Raphson method effectively and highly precisely approximates solutions for different ODEs. Some examples, however, show differences between numerical and analytical answers, suggesting possible problems with error accumulation or inherent constraints of the approach. Problem difficulty, step size, and initial guesses all affect convergence. Although the Newton-Raphson approach solves ODEs numerically quite well, it must be carefully validated against analytical solutions. The performance of the procedure depends on elements particular to the problem that must be taken into account in application. The need for choosing suitable numerical methods for solving ODEs in scientific and technical domains is underlined by this work. The results guide future research and useful implementations by offering an understanding of the strengths and constraints of Newton-Raphson-based solvers.

Numerical Solution of 2D Nonlinear Volterra-Fredholm Integral Equations using Polynomial Collocation Method

2025-04-02
Albert A. Shalangwa , M. R. Odekunle , Solomon Ortwer Adee
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 ā€¦ 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