Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations

This thesis considers linear Volterra-Fredholm integral equations (lEs) and linear Volterra-Fredholm integro-differential equation (IDEs) of order one, two and high order, with boundary condition. Polynomial approximation method together with Gauss- Legendre quadrature formula (QF) are proposed t...

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Main Author:	Massamdi bin Kammuji
Format:	Thesis
Language:	English
Subjects:	Chemistry Physical and theoretical Physical chemistry linear Volterra-Fredholm integral equations (lEs) linear Volterra-Fredholm integro-differential equation (IDEs)
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spelling	my-usim-ddms-124462024-05-29T04:11:23Z Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations Massamdi bin Kammuji This thesis considers linear Volterra-Fredholm integral equations (lEs) and linear Volterra-Fredholm integro-differential equation (IDEs) of order one, two and high order, with boundary condition. Polynomial approximation method together with Gauss- Legendre quadrature formula (QF) are proposed to find the approximate solution of those equations. The Legendre truncated series of order n is chosen as the basis function to estimate the solution. Gauss-Legendre QF and the collocation method are applied to reduce the Volterra-Fredholm lEs into the system of linear algebraic equation. The later equation is then solved by Gauss elimination method. For the first and second order Volterra-Fredholm IDEs, reduction method is used before applying the polynomial approximation method with quadrature forniula. A different approach is used for the high order Volterra-Fredholm IDEs. The operational matrix of derivative for Legendre polynomials is used to transform the high order Volterra-Fredholm IDEs into matrix equation. Gauss-Legendre QF and the collocation method are used to form a system of linear algebraic equation. The system is then solved by Gauss elimination method. For the first and second order Volterra-Fredholm IDEs, convergence analysis and rate of convergence of the proposed method are obtained in the class of smooth functions C' [u, h] with the Chebyshev none. Test problems are provided to show the accuracy and validity of the proposed method. The results show that the error of the proposed method decreased faster as n increases accept for the equation that has 4.5 in its exact solution and the equation that appear many derivatives of unknown function. Some test problems show that the proposed method able to solve exactly the problems when the solution is the polynomial form. The errors for some test problems are compared with errors obtained by other methods. The operational matrix of derivative for Legendre polynomials solved the problems without reduce it into integral equation and make fewer jobs. All the numerical results are obtained using Maple 17. Universiti Sains Islam Malaysian 2018-03 Thesis en https://oarep.usim.edu.my/handle/123456789/12446 https://oarep.usim.edu.my/bitstreams/4f2caf38-681e-4f52-8d85-226604d7b522/download 8a4605be74aa9ea9d79846c1fba20a33 https://oarep.usim.edu.my/bitstreams/913f9991-bac2-4e4d-a152-854440f0cf94/download a279404e4c622010f158be8d2b359e83 https://oarep.usim.edu.my/bitstreams/b7ffe084-a696-49f3-bdae-535dc15badc1/download 0b0d3fd6946476a040f76e6a8c97c18d https://oarep.usim.edu.my/bitstreams/f95f71b5-244e-4d23-9c70-0d18bff1470a/download 8168c7a391ca3c0e32bcf1a4140859ff https://oarep.usim.edu.my/bitstreams/5de38dd9-a313-4d7d-9166-c6423c798730/download cd52d58a29b1770598498df72990eea4 https://oarep.usim.edu.my/bitstreams/a2dc59b0-6f07-42a2-94df-414fde76fe9d/download afef89a5a60673c06dec1b4312cfeb1f https://oarep.usim.edu.my/bitstreams/43ca1614-3de7-41cf-a319-ed7c51531329/download dad9e293bb81ea7a523a00c9dd754e03 https://oarep.usim.edu.my/bitstreams/04bb90cc-c6bd-4fda-83ae-7c6108238790/download de04ec604418efb9b886da59ff2ec21c https://oarep.usim.edu.my/bitstreams/df793c87-e20b-47c1-9ded-474eaf665100/download a0fcd9bd82d1a3d7b0bb3907b89db798 https://oarep.usim.edu.my/bitstreams/e86272f0-cd1a-4e0f-ae45-3effd22bbdb2/download 4db4003d97db0be31a81aa48d2655570 https://oarep.usim.edu.my/bitstreams/5a90bb3f-8c4a-4d07-9541-0a84c8db0df6/download 5e695462526671515c1635945fcd17fe Chemistry Physical and theoretical Physical chemistry linear Volterra-Fredholm integral equations (lEs) linear Volterra-Fredholm integro-differential equation (IDEs)
institution	Universiti Sains Islam Malaysia
collection	USIM Institutional Repository
language	English
topic	Chemistry Physical and theoretical Physical chemistry linear Volterra-Fredholm integral equations (lEs) linear Volterra-Fredholm integro-differential equation (IDEs)
spellingShingle	Chemistry Physical and theoretical Physical chemistry linear Volterra-Fredholm integral equations (lEs) linear Volterra-Fredholm integro-differential equation (IDEs) Massamdi bin Kammuji Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
description	This thesis considers linear Volterra-Fredholm integral equations (lEs) and linear Volterra-Fredholm integro-differential equation (IDEs) of order one, two and high order, with boundary condition. Polynomial approximation method together with Gauss- Legendre quadrature formula (QF) are proposed to find the approximate solution of those equations. The Legendre truncated series of order n is chosen as the basis function to estimate the solution. Gauss-Legendre QF and the collocation method are applied to reduce the Volterra-Fredholm lEs into the system of linear algebraic equation. The later equation is then solved by Gauss elimination method. For the first and second order Volterra-Fredholm IDEs, reduction method is used before applying the polynomial approximation method with quadrature forniula. A different approach is used for the high order Volterra-Fredholm IDEs. The operational matrix of derivative for Legendre polynomials is used to transform the high order Volterra-Fredholm IDEs into matrix equation. Gauss-Legendre QF and the collocation method are used to form a system of linear algebraic equation. The system is then solved by Gauss elimination method. For the first and second order Volterra-Fredholm IDEs, convergence analysis and rate of convergence of the proposed method are obtained in the class of smooth functions C' [u, h] with the Chebyshev none. Test problems are provided to show the accuracy and validity of the proposed method. The results show that the error of the proposed method decreased faster as n increases accept for the equation that has 4.5 in its exact solution and the equation that appear many derivatives of unknown function. Some test problems show that the proposed method able to solve exactly the problems when the solution is the polynomial form. The errors for some test problems are compared with errors obtained by other methods. The operational matrix of derivative for Legendre polynomials solved the problems without reduce it into integral equation and make fewer jobs. All the numerical results are obtained using Maple 17.
format	Thesis
author	Massamdi bin Kammuji
author_facet	Massamdi bin Kammuji
author_sort	Massamdi bin Kammuji
title	Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
title_short	Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
title_full	Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
title_fullStr	Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
title_full_unstemmed	Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
title_sort	polynomial and quadrature method for solving linear integraland integro-differential equations
granting_institution	Universiti Sains Islam Malaysian
_version_	1812444715670306816

Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations

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