Abstract
Local polynomial regression (LPR) is applied to solve the partial differential equations (PDEs). Usually, the solutions of the problems are separation of variables and eigenfunction expansion methods, so we are rarely able to find analytical solutions. Consequently, we must try to find numerical solutions. In this paper, two test problems are considered for the numerical illustration of the method. Comparisons are made between the exact solutions and the results of the LPR. The results of applying this theory to the PDEs reveal that LPR method possesses very high accuracy, adaptability, and efficiency; more importantly, numerical illustrations indicate that the new method is much more efficient than B‐splines and AGE methods derived for the same purpose.
Highlights
There are some effective and convenient numerical methods for partial differential equations problems with initial and boundary values;for example, the radial basis functions are used to solve two-dimensional sine-Gordon equation in 1, a family of second-order methods are used to solve variable coefficient fourth-order parabolic partial differential equations in 2, and fifth-degree B-spline solution is available for a fourth-order parabolic partial differential equations in 3
Among n−1 bandwidth, the bandwidth that makes eMS minimized is the optimal bandwidth. Another issue in multivariate local polynomial fitting is the choice of the order of the polynomial
LPR method can be applied for the numerical solution of some kinds of PDEs
Summary
There are some effective and convenient numerical methods for partial differential equations problems with initial and boundary values;for example, the radial basis functions are used to solve two-dimensional sine-Gordon equation in 1 , a family of second-order methods are used to solve variable coefficient fourth-order parabolic partial differential equations in 2 , and fifth-degree B-spline solution is available for a fourth-order parabolic partial differential equations in 3. Caglar 4 have used local polynomial regression LPR method for the numerical solution of fifth-order boundary value problems 4. They manage to solve linear Fredholm and Volterra integral equations by using LPR method 5. We can consider LPR method is applied to some PDEs with initial and boundary values, and numerical results demonstrate that local polynomial. We consider the numerical approximation for the simple nonhomogeneous PDE with initial values with the use of local polynomial regression:.
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