An efficient method of approximate particular solutions using polynomial basis functions. (English) Zbl 1464.65228
Summary: The most challenging task of the method of approximate particular solutions (MAPS) is the generation of the closed-form particular solutions with respect to the given differential operator using various basis functions. These particular solutions have to be generated prior to the solution process of the partial differential equations. In this paper, we propose a different approach without the tedious and inefficient solution procedure using symbolic computation to produce the closed-form particular solutions. The proposed approach is introduced and extended to solve a large class of elliptic partial differential equations (PDEs) based on the method of approximate particular solutions (MAPS).
Numerical results show the proposed approach is simple, efficient, accurate, and stable. Five different numerical examples are presented to demonstrate the effectiveness of the proposed method.
Numerical results show the proposed approach is simple, efficient, accurate, and stable. Five different numerical examples are presented to demonstrate the effectiveness of the proposed method.
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
method of approximate particular solutions; Helmholtz-type equation; polynomial basis functions; multiple scale techniqueReferences:
| [1] | Chen, C. S.; Fan, C. M.; Wen, P. H., The method of particular solutions for solving elliptic problems with variable coefficients, Int J Comput Methods, 8, 545-559 (2011) · Zbl 1245.65172 |
| [2] | Chen, C. S.; Fan, C. M.; Wen, P. H., The method of particular solutions for solving certain partial differential equations, Numer Methods Partial Differ Equ, 28, 506-522 (2012) · Zbl 1242.65267 |
| [3] | Chen, W.; Fu, Z.-J.; Chen, C. S., Recent advances on radial basis function collocation methods (2014), Springer · Zbl 1282.65160 |
| [4] | Chang, W.; Chen, C. S.; Li, W., Solving fourth order differential equations using particular solutions of Helmholtz-type equations, Appl Math Lett, 86, 179-185 (2018) · Zbl 1416.65470 |
| [5] | Cheng, A. H.D., Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions, Eng Anal Bound Elem, 24, 531-538 (2000) · Zbl 0966.65088 |
| [6] | Dangal, T.; Chen, C. S.; Lin, J., Polynomial particular solution for solving elliptic partial differential equations, Comput Math Appl, 73, 60-70 (2017) · Zbl 1368.65255 |
| [7] | Ding, J.; Chen, C. S., Particular solutions of some elliptic partial differential equations via recursive formulas, J Univ Sci Technol China, 37, 11, 1338-1347 (2007) |
| [8] | Dou, F. F.; Hon, Y. C., Fundamental kernel-based method for backward space-time fractional diffusion problem, Comput Math Appl, 71, 356-367 (2016) · Zbl 1443.65174 |
| [9] | Dou, F.; Liu, Y.; Chen, C. S., The method of particular solutions for solving nonlinear poisson problems, Comput Math Appl, 77, 2, 501-513 (2019) · Zbl 1442.35071 |
| [10] | Fan, C. M.; Yang, C. H.; Lai, W. S., Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solution, Eng Anal Bound Elem, 57, 47-57 (2015) · Zbl 1403.78028 |
| [11] | Fausshauer, G. E., Solving partial differential equations by collocation with radial basis functions, (Mehaute, A. L.; Rabut, C.; Schmaker, L. L., Surface fitting and multiresolution methods (1997), Vanderblt University Press: Vanderblt University Press Nashville, TN), 131-138 · Zbl 0938.65140 |
| [12] | Golberg, M. A.; Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (Golberg, M. A., Boundary integral methods: numerical and mathematical aspects (1998), WIT Press), 103-176 · Zbl 0945.65130 |
| [13] | Golberg, M. A.; Muleshkov, A. S.; Chen, C. S.; Cheng, A. H.D., Polynomial particular solutions for certain kind of partial differential operators, Numer Methods Partial Differ Equ, 19, 112-133 (2003) · Zbl 1019.65096 |
| [14] | Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput Math Appl, 19, 147-161 (1990) · Zbl 0850.76048 |
| [15] | Le, J.; Jin, H.; Lv, X. G.; Cheng, Q. S., A preconditioned method for the solution of the robbins problem for the Helmholtz equation, Anziam J, 52, 87-100 (2010) · Zbl 1234.65037 |
| [16] | Li, H. B.; Huang, T. Z.; Zhang, Y.; Liu, X. P.; Gu, T. X., Chebyshev-type methods and preconditioning techniques, Appl Math Comput, 218, 260-270 (2011) · Zbl 1226.65024 |
| [17] | Lin, J.; Chen, C. S.; Wang, F.; Dangal, T., Method of particular solutions using polynomial basis functions for the simulation of plate bending vibrations problems, Appl Math Model, 49, 452-469 (2017) · Zbl 1480.74115 |
| [18] | Lin, J.; Zhang, C. Z.; Sun, L. L.; Lu, J., Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method, Adv Appl Math Mech, 10, 2, 322-342 (2018) · Zbl 1488.65719 |
| [19] | Liu, C., A multiple-scale trefftz method for an incomplete cauchy problem of biharmonic equation, Eng Anal Bound Elem, 37, 1445-1456 (2013) · Zbl 1287.65103 |
| [20] | Lu, J.; Yu, H.; Lin, J.; Dangal, T., Polynomial particular solutions for the solutions of PDEs with variables coefficients, Adv Appl Math Mech, 10, 5, 1-14 (2018) |
| [21] | Muleshkov, A. S.; Golberg, M. A.; Chen, C. S., Particular solutions of helmholtz-type operators using higher order polyhrmonic splines, Comput Mech, 23, 5-6, 411-419 (1999) · Zbl 0938.65139 |
| [22] | Muleshkov, A. S.; Golberg, M. A.; Chen, C. S., Particular solutions for axisymmetric Helmholtz-type operators, Eng Anal Bound Elem, 29, 1066-1076 (2005) · Zbl 1182.76930 |
| [23] | Ngo-Cong, D.; Tran, C.-D.; Mai-Duy, N.; Tran-Cong, T., Imcompressible smoothed particle hydrodynamics-moving IRBFN method for viscous flow problems, Eng Anal Bound Elem, 59, 172-186 (2015) · Zbl 1403.76168 |
| [24] | Shankar, V.; Wright, G. B.; Kirby, R. M.; Fogelson, A. L., A radial basis function (RBF)-finite difference (FD) method for diffuson and reaction-diffusion equation on surfaces, J Sci Comput, 63, 745-768 (2015) · Zbl 1319.65079 |
| [25] | Tsai, C. C., Particular solutions of Chebyshev polynomials for polyharmonic and poly-Helmholtz equations, Comput Model Eng Sci, 27, 3, 151-162 (2008) · Zbl 1232.65173 |
| [26] | Tsai, C. C.; Chen, C. S.; Hsu, T. W., The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations, Eng Anal Bound Elem, 33, 12, 1396-1402 (2009) · Zbl 1244.65225 |
| [27] | Tsai, C. C.; Cheng, A. H.-. D.; Chen, C. S., Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators, Eng Anal Bound Elem, 33, 514-521 (2009) · Zbl 1244.65209 |
| [28] | Wang, C.; Huang, T. Z.; Wen, C., A new preconditioner for indefinite and asymmetric matrices, Appl Math Comput, 219, 11036-11043 (2013) · Zbl 1302.65073 |
| [29] | Wang, M.; Watson, D.; Li, M., The method of particular solutions with polynomial basis functions for solving axisymmetric problems, Eng Anal Bound Elem, 90, 39-46 (2018) · Zbl 1403.65262 |
| [30] | Wang, D.; Chen, C. S.; Fan, C. M., The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy-Navier equations, Math Comput Simul, 159, 119-135 (2019) · Zbl 1540.65510 |
| [31] | Wei, X.; Sun, L.; Yin, S.; Chen, B., A boundary-only treatment by singular boundary method for two-dimensional inhomongeneous problems, Appl Math Model, 62, 338-351 (2018) · Zbl 1460.65151 |
| [32] | Yao, G.; Chen, C. S.; Tsai, C. C., A revisit on the derivation of the particular solution for the differential operator \(δ^2\) ± \(λ^2\), Adv Appl Math Mech, 1, 6, 750-768 (2009) · Zbl 1262.35086 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
