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

S.Yu. Reutskiy1

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 ∈ Rd, d =1,2,3. We assume that the nonlinear term can be represented as a linear combination of basis functions F(u,x) = ∑mMqmφm. We use the basis functions φm of three types: the the monomials, the trigonometric functions and the multiquadric radial basis functions. For basis functions φm of each kind there exist particular solutions of the equation 2ϕm = φm in an analytic form. This permits to write the approximate solution in the form uM = uf +∑mMqmΦm, where Φm = ϕm + ωm. The term ωm provides that Φm satisfies the homogeneous conditions on the boundary of the domain. Substituting uM into the equation for F, we transform it to the system of nonlinear equations F(uM,xn) = ∑mMqmφm(xn), n = 1,...,M for the unknown coefficients qm. 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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APA Style
Reutskiy, S. (2012). A novel method for solving one-, two- and three-dimensional problems with nonlinear equation of the poisson type. Computer Modeling in Engineering & Sciences, 87(4), 355-386. https://doi.org/10.3970/cmes.2012.087.355
Vancouver Style
Reutskiy S. A novel method for solving one-, two- and three-dimensional problems with nonlinear equation of the poisson type. Comput Model Eng Sci. 2012;87(4):355-386 https://doi.org/10.3970/cmes.2012.087.355
IEEE Style
S. Reutskiy, "A Novel Method for Solving One-, Two- and Three-Dimensional Problems with Nonlinear Equation of the Poisson Type," Comput. Model. Eng. Sci., vol. 87, no. 4, pp. 355-386. 2012. https://doi.org/10.3970/cmes.2012.087.355



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