On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

Assari, Pouria; Asadi‐Mehregan, Fatemeh; Dehghan, Mehdi

doi:10.1080/00207160.2018.1500693

Cited by 24 publications

(15 citation statements)

References 62 publications

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“…In particular, meshless discrete Galerkin methods were successfully developed for solving Fredholm and Hammerstein integral equations for various bases. See, for example, [16] for an effective and stable method to estimate the solution to Hammerstein integral equations with free shape parameter radial basis functions, constructed on scattered points; [17,18], for effective computational meshless methods for solving Fredholm integral equations of the second kind with logarithmic and weakly singular kernels, using radial basis functions, meshless product integration and collocation methods; and [19,20], for efficient meshless methods for solving non-linear weakly singular Fredholm integral equations, combining discrete collocation method with locally supported radial basis functions and thin-plate splines.…”

Section: Discussionmentioning

confidence: 99%

Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

Seminara

Troparevsky

2018

MCA

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In this work we obtain approximate solutions for Fredholm integral equations of the second kind by means of Petrov–Galerkin method, choosing “regular pairs” of subspaces, Xn,Yn, which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvability and numerical stability of the approximation scheme in an easy way, and the selection of orthogonal basis for the subspaces make the calculations quite simple. Afterwards, we explore an interesting phenomenon called “superconvergence”, observed in the 1970s by Sloan: once the approximations un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. We illustrate both procedures of approximation by means of two numerical examples: one for a continuous kernel, and the other for a weakly singular one.

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Section: Discussionmentioning

confidence: 99%

Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

Seminara

Troparevsky

2018

MCA

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“…For tests, we apply the local Gaussian (LGA), local multiquadric (LMQ) and local inverse multiquadrics (LIMQ) which the accuracy and stability of them heavily depend on the shape parameters

c > 0

. Since LRBFs have much more freedom in choosing the shape parameter, we select middle values

c = 0.1 \times \sqrt{N}

(

c = 0.2 \times \sqrt{N}

) for LGAs and

c = \frac{5}{\sqrt{N}}

(

c = \frac{10}{\sqrt{N}}

) for LMQs and LIMQs in one‐dimensional (two‐dimensional) case . In illustrative examples, we employ 5‐points composite Gauss‐Legendre quadrature rule with

M = 5

for approximating integrals in the scheme.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In the expansion (12), the coefficients { } ∈ are determined by enforcing the interpolation conditions…”

Section: Locally Supported Rbfsmentioning

confidence: 99%

“…Theorem 5.1 [12]. Suppose that Φ is positive definite RBF with infinite smoothness and ∈ ℵ Φ [ , ] .…”

Section: Error Analysismentioning

confidence: 99%

“…LGAs and = 5 √ ( = 10 √ ) for LMQs and LIMQs in one-dimensional (two-dimensional) case. [12,26] In illustrative examples, we employ 5-points composite Gauss-Legendre quadrature rule with = 5 for approximating integrals in the scheme. The cover size of the node ∈ ℝ is determined as follows [41]:…”

Section: Numerical Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

Assari

Asadi‐Mehregan

2019

Z Angew Math Mech

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Integro‐differential equations with non‐integer order derivatives are an all‐purpose subdivision of fractional calculus. In the current paper, we present a numerical method for solving fractional Volterra‐Fredholm integro‐differential equations of the second kind. To establish the scheme, we first convert these types of integro‐differential equations to mixed integral equations by fractional integrating from both sides of them. Then, the discrete collocation method by combining the locally supported radial basis functions are used to approximate the mentioned integral equations. Since the local method proposed in this paper estimates an unknown function via a small set of data instead of all points in the solution domain, it uses much less computer memory in comparison with the global cases. We apply the nonuniform Gauss‐Legendre integration rule to compute the singular‐fractional integrals appearing in the scheme. As the scheme does not need any background meshes, it can be recognized as a meshless method. Numerical results are included to show the validity and efficiency of the new technique and confirm that the algorithm of the presented approach is attractive and easy to implement on computers.

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Solving Integral Equations by LS-SVR

Parand

Aghaei

Jani

et al. 2023

Industrial and Applied Mathematics

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On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

Cited by 24 publications

References 62 publications

Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

Solving Integral Equations by LS-SVR

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