A meshless local discrete collocation (MLDC) scheme for solving 2‐dimensional singular integral equations with logarithmic kernels

Abstract

In the current investigation, we study a numerical technique to solve 2‐dimensional singular integral equations of the second kind with logarithmic kernels. These integral equations are a general form of the problem of obtaining the cross‐sectional separation of current in an infinitely long thin filament of conduct bar which arises in electrical engineering. The moving least squares scheme estimates a function without mesh generation on the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation method in addition to the shape functions of moving least squares established on scattered points is used to approximate the solution of integral equations. To compute logarithm‐like singular integrals appeared in the process of the scheme, we use a particular dual nonuniform composite Gauss‐Legendre integration rule on normal domains. The proposed scheme does not require any meshes, so it is meshless and does not depend to the domain form. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

“…• Meshless techniques based on collocation schemes (strong forms), such as the meshless collocation method based on radial basis functions (RBFs) (Parand et al 2011;Kansa 1990;Jakobsson et al 2009;Abbasbandy et al 2012Abbasbandy et al , 2013Kamranian et al 2016;Moradipour and Yousefi 2018). • Meshless techniques based on the combination of weak forms and collocation method (Santin and Schaback 2016;Schaback 2015;Young et al 2008;Assari and Dehghan 2018;Shirzadi 2010, 2011;Dehghan and Ghesmati 2010;Liu et al 2002;Liu and Gu 2001;Shivanian 2013Shivanian , 2014Khodabandehlo 2014, 2016;Hosseini et al 2015Hosseini et al , 2016.…”

Meshless approximation method of one-dimensional oscillatory Fredholm integral equations

“…As from (17), it is clear that we need only the first and the last entry of a vector having zero entries in between, therefore we need a vector of the form:…”

“…The global meshless method is constructed on undertaking the whole domain into account, has some drawbacks, like dense ill-conditioning and sensitivity of shape parameter. In order to avert the bad effects of the global method, a local meshless method has been suggested by the researchers [1,2,4,17,21,22]. Local meshless methods are comparatively slightly insensitive to the variation of the shape parameter than the global counterparts.…”

Formulation of pseudospectral meshless radial point Hermit interpolation for the Motz problem and comparison to pseudospectral meshless radial point interpolation

“…Meshless techniques based on collocation schemes (strong forms), such as the meshless collocation method based on RBFs (Kansa, 1990; Jakobsson et al , 2009; Abbasbandy et al , 2012, 2013; Kamranian et al , 2017; Abbaszadeh et al , 2019; Esmaeilbeigi et al , 2017a, b; Mirzaee and Samadyar, 2018, 2019; Samadyar and Mirzaee, 2019). Meshless techniques based on the combination of weak forms and collocation method (Assari and Dehghan, 2018; Shivanian, 2013; Abbasbandy and Shirzadi, 2010, 2011; Dehghan and Ghesmati, 2010; Liu et al , 2002; Liu and Gu, 2001; Shivanian, 2014, 2015a, 2016b; Shivanian and Khodabandehlo, 2014, 2016; Shivanian et al , 2016; Hosseini et al , 2015, 2016; Shivanian and Khodayari, 2017). …”