Abstract
We extended the moving least squares (MLS) and modified moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations. This modification is proposed on the quadratic base functions $$(m=2)$$ by imposing additional terms based on the coefficients of the polynomial base functions. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Additionally, finding the optimum value for the radius of the domain influence is an open problem for MLS-based methods. So an efficient algorithm is introduced for computing a suitable value of dilatation parameter to determine the radius of the support domain. This algorithm able to prevent the singular matrix which is an outcome of adverse selection of the radius of influence domain. In numerical examples are provided to enable us to compare MMLS method and standard MLS method by the new proposed algorithm. Comparing the errors of MMLS and MLS method determines the capability and accuracy of applied techniques to solve systems of integral equation problems. This indicates the advantage of the proposed method respect to MLS method.
Highlights
In mathematics, there are many functional equations of the description of a real system in the natural sciences and disciplines of engineering
The Moving Least Square (MLS) method is a feasible numerical approximation method that is an extension of the least squares method, it is the component of the class of meshless schemes that have a highly accurate approximation
Two meshless techniques called moving least squares and modified Moving least-squares approximation are applied for solving the system of functional equations
Summary
There are many functional equations of the description of a real system in the natural sciences (such as physics, biology, Earth science, meteorology) and disciplines of engineering. We can point to some mathematical model from physics that describe heat as a partial differential equation and the inverse problem of it’s as integro-differential equations. Another example in nature is Laplace’s equation which corresponds to the construction of potential for a vector field whose effect is known at the boundary of Domain alone. There are many significant analytical methods for solving integral equations but most of them especially in nonlinear cases, finding an analytical representation of the solution is so difficult, it is required to obtain approximate solutions. The interested reader can find several numerical methods for approximating the solution of these problems in [5,6,7,8,9,10,11,12,13,14] and the references therein
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