zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method. (English) Zbl 1254.65138
Summary: We present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated.

65T60Wavelets (numerical methods)
Full Text: DOI
[1] Mittal, R. C.; Gahlaut, S.: High-order finite-differences schemes to solve Poisson’s equation in polar coordinates, IMA J. Numer. anal. 11, 261-270 (1987) · Zbl 0732.65097 · doi:10.1093/imanum/11.2.261
[2] Perrey-Debain, E.; Morsche, H. G. Ter: B-spline approximation and fast wavelet transform for an efficient evaluation of particular solutions for Poisson’s equation, Eng. anal. Bound. elem. 26, No. 1, 1-13 (2002) · Zbl 0996.65130 · doi:10.1016/S0955-7997(01)00080-7
[3] Sutmann, Godehard; Steffen, Bernhard: High-order compact solvers for the three-dimensional Poisson equation, J. comput. Appl. math. 187, 142-170 (2006) · Zbl 1081.65099 · doi:10.1016/j.cam.2005.03.041
[4] Ge, Yongbin: Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation, J. comput. Phys. 229, 6381-6391 (2010) · Zbl 1197.65169 · doi:10.1016/j.jcp.2010.04.048
[5] Gumerov, Nail A.; Duraiswami, Ramani: Fast multipole method for the biharmonic equation in three dimensions, J. comp. Phys. 215, 363-383 (2006) · Zbl 1103.65122 · doi:10.1016/j.jcp.2005.10.029
[6] Khattar, Dinesh; Singh, Swarn; Mohanty, R. K.: A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations, Appl. math. Comput. 215, 3036-3044 (2009) · Zbl 1180.65137 · doi:10.1016/j.amc.2009.09.052
[7] Altas, Irfan; Erhel, Jocelyne; Gupta, Murli M.: High accuracy solution of three-dimensional biharmonic equations, Numer. algorithms 29, 1-19 (2002) · Zbl 0992.65115 · doi:10.1023/A:1014866618680
[8] Jeon, Youngmok: New indirect scalar boundary integral equation formulas for the biharmonic equation, J. comput. Appl. math. 135, No. 2, 313-324 (2001) · Zbl 0986.31003 · doi:10.1016/S0377-0427(00)00590-2
[9] Mai-Duy, N.; Tanner, R. I.: A spectral collocation method based on integrated Chebyshev polynomials for two-dimensional biharmonic boundary-value problems, J. comput. Appl.math. 201, 30-47 (2007) · Zbl 1110.65112 · doi:10.1016/j.cam.2006.01.030
[10] Mai-Duy, N.; Tran-Cong, T.: Solving biharmonic problems with scattered-point discretization using indirect radial-basis-function networks, Eng. anal. Bound. elem. 30, No. 2, 77-87 (2006) · Zbl 1195.65179 · doi:10.1016/j.enganabound.2005.10.004
[11] Mai-Duy, N.; Tanner, R. I.: An effective high order interpolation scheme in BIEM for biharmonic boundary value problems, Eng anal bound elem 29, No. 3, 210-223 (2005) · Zbl 1182.74226 · doi:10.1016/j.enganabound.2005.01.005
[12] Li, Xiaolin; Zhu, Jialin: A Galerkin boundary node method for biharmonic problems, Eng anal bound elem 33, No. 6, 858-865 (2009) · Zbl 1244.65175
[13] Mai-Duy, N.; See, H.; Tran-Cong, T.: A spectral collocation technique based on integrated Chebyshev polynomials for biharmonic problems in irregular domains, Appl math model 33, No. 1, 284-299 (2009) · Zbl 1167.65454 · doi:10.1016/j.apm.2007.11.002
[14] Dissanayake, M. W. M.G.; Phan-Thien, N.: Neural-network-based approximations for solving partial differential equations, Commun. numer. Methods eng. 10, No. 3, 195-201 (1994) · Zbl 0802.65102 · doi:10.1002/cnm.1640100303
[15] Mai-Duy, N.; Tanner, R. I.: A collocation method based on one-dimensional RBF interpolation scheme for solving pdes, Int. J. Numer. methods heat fluid flow 17, No. 2, 165-186 (2007) · Zbl 1231.76188 · doi:10.1108/09615530710723948
[16] Lepik, Ü.: Haar wavelet method for solving higher order differential equation, Int. J. Math. comput. 1, No. 8, 84-94 (2008)
[17] Lepik, Ü.: Solving integral and differential equations by the aid of non-uniform Haar wavelets, Appl. math. Comput. 198, No. 1, 326-332 (2008) · Zbl 1137.65071 · doi:10.1016/j.amc.2007.08.036
[18] Lepik, Ü.: Solving fractional integral equations by the Haar wavelet method, Appl. math. Comput. 214, No. 2, 468-478 (2009) · Zbl 1170.65106 · doi:10.1016/j.amc.2009.04.015