Open Access

ARTICLE

A Novel Method for Solving One-, Two- and Three-Dimensional Problems with Nonlinear Equation of the Poisson Type

by ,

1 Science and Technology Center of Magnetism of Technical Objects. The National Academy of Science of Ukraine, Industrialnaya St.,19, 61106, Kharkov, Ukraine

Computer Modeling in Engineering & Sciences 2012, 87(4), 355-386. https://doi.org/10.3970/cmes.2012.087.355

Abstract

The paper presents a new meshless numerical technique for solving nonlinear Poisson-type equation ∇²u = f (x) + F(u,x) for x ∈ R^d, d =1,2,3. We assume that the nonlinear term can be represented as a linear combination of basis functions F(u,x) = ∑_m^Mq_mφ_m. We use the basis functions φ_m of three types: the the monomials, the trigonometric functions and the multiquadric radial basis functions. For basis functions φ_m of each kind there exist particular solutions of the equation ∇²ϕ_m = φ_m in an analytic form. This permits to write the approximate solution in the form u_M = u_f +∑_m^Mq_mΦ_m, where Φ_m = ϕ_m + ω_m. The term ω_m provides that Φ_m satisfies the homogeneous conditions on the boundary of the domain. Substituting u_M into the equation for F, we transform it to the system of nonlinear equations F(u_M,x_n) = ∑_m^Mq_mφ_m(x_n), n = 1,...,M for the unknown coefficients q_m. Then the nonlinear system is solved numerically. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate and efficient in dealing with complicated geometry and strong nonlinearity.

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