Solving Fredholm integral equations of the first kind using Müntz wavelets

Bahmanpour, Maryam; Kajani, M. Tavassoli; Maleki, Mohammad

doi:10.1016/j.apnum.2019.04.007

Cited by 21 publications

(9 citation statements)

References 15 publications

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“…In [75], a numerical direct method based on hybrid Block-Pulse functions and Legendre polynomials is proposed to solve Fredholm integral equation of the first kind. Bahmanpour et al [77] introduce Müntz wavelets by using the Müntz-Legendre polynomials on the interval [0, 1]. Xie [78] studied the first kind of Fredholm integral equation with perturbation of original data under truncation strategy and obtained the fast solution of discrete regularization equation.…”

Section: Multilevel Iteration Methodsmentioning

confidence: 99%

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An overview of numerical methods for the first kind Fredholm integral equation

Zhang

2019

SN Appl. Sci.

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In the field of engineering technology, many problems can be transformed into the first kind Fredholm integral equation, which has a prominent feature called "ill-posedness". This property makes it difficult to find the analytical solution of first kind Fredholm integral equation. Therefore, how to find the numerical solution of first kind Fredholm integral equation has been a common concern of domestic and overseas scholars in recent years. In this article, various numerical solution methods of first kind Fredholm integral equation are introduced in detail. First, the existence and convergence of the solution of the integral equation are given. Second, the current mainstream numerical methods, such as regularization method, wavelet analysis method and multilevel iteration method are introduced in detail. Finally, we presented a concise overview of the numerical method of first kind Fredholm integral equation.

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Section: Multilevel Iteration Methodsmentioning

confidence: 99%

“…If the intelligent algorithm is used to solve the discrete algebraic equations of the integral equation, the ideal result may be obtained. [105] Bahmanpour et al [77] Adibi and Assari [65] Maleknejad and Saeedipoor [75] Maleknejad and Sohrabi [61] Max. absolute error 2.09e−5 4.21e−5 8.92e−4 9.01e−4 8.54e−4…”

Section: Conclusion and Prospectsmentioning

confidence: 99%

An overview of numerical methods for the first kind Fredholm integral equation

Zhang

2019

SN Appl. Sci.

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“…1); t is the normalized time and m is the degree of Muntz polynomials. The definition of Muntz wavelets on the interval ½0; 1Þ for a given 0 < γ < 1 is as follows (Bahmanpour et al, 2018(Bahmanpour et al, , 2019…”

Section: Muntz Waveletsmentioning

confidence: 99%

“…The dilation parameter is 2 À(kÀ1) , the translation parameter is (nÀ1) 2 À(kÀ1) , and the coefficient ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmγ þ ð1=2ÞÞ p is for orthonormality. Here, P m (t) are the Muntz polynomials of order m. These polynomials are orthogonal under the weight function w(t) = 1 on the interval ½0; 1Þ and satisfying the following recursive relation (Bahmanpour et al, 2019) P 0 ðt, γÞ ¼ 1 (6)…”

Section: Muntz Waveletsmentioning

confidence: 99%

Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets

Rayal

Verma

2020

Journal of Vibration and Control

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In this study, an approximation method with an integral operational matrix based on the Muntz wavelets basis is presented to solve the variational problems of moving or fixed boundary conditions and a computational algorithm is given for the suggested approach. First, the integral operational matrix is created through the Muntz wavelets. Then, by using this integral operational matrix with Lagrange multipliers, the present approach reduces the variational problem into the system of algebraic equations. This approach is examined by some illustrative examples, and the acquired results prove that the suggested approach can solve the variational problems effectively with higher accuracy. The proposed approach yields better and comparable results with some other existing schemes given in the literature. The approximate wavelet solutions derived by the suggested approach are very identical to the corresponding exact solution.

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“…Mokhtary et al (2016) studied the operational tau methods based on M-L polynomials for solving FDEs. Also, in Bahmanpour et al (2019), Müntz wavelet combination with a matrix method was introduced to solve Fredholm integral equations of the first kind. Shen and Wang (2016) develop a Müntz-Galerkin method to deal with the singular behaviors of the mixed Dirichlet-Neumann boundary value problems and obtained optimal error estimates.…”

Section: Introductionmentioning

confidence: 99%

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

2020

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The main purpose of this work is to develop effective spectral collocation methods for solving numerically general form of non-linear fractional two-point boundary value problems with two end-point singularities. Specifically, we define two classes of Müntz-Legendre polynomials to deal with the singular behaviors of the solutions at both end-points, which enhance greatly the accuracy of the numerical solutions. Important and practical formulas for ordinary derivatives of Müntz-Legendre polynomials are derived. Then using a three-term recurrence formula and Jacobi-Gauss quadrature rules, applicable and stable numerical approaches for computing left and right Caputo fractional derivatives of these basis functions are provided. One of the important features of the present method is to show how to recover spectral accuracy from the end-point singularities for the coupled systems of fractional differential equations with left and right fractional derivatives using the mentioned approximation basis. Moreover, we present a detailed analysis with accurate convergence rates of the singular behavior of the solution to a rather general system of nonlinear fractional differential equations including left and right fractional derivatives. Numerical results show the expected convergence rates. Finally, some numerical examples are given to demonstrate the performance of the proposed schemes.

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Solving Fredholm integral equations of the first kind using Müntz wavelets

Cited by 21 publications

References 15 publications

An overview of numerical methods for the first kind Fredholm integral equation

An overview of numerical methods for the first kind Fredholm integral equation

Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

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