Geodesic
Convergence of Fourier method connected with orthogonal splines
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 3, pp. 245-259.

Voir la notice de l'article provenant de la source Math-Net.Ru

Fourier method and Fourier series have wide fields of application. The use of the theory of orthogonal splines, created by the author of this article and developed in the last thirty years, has led to significant progress in a number of numerical and analytical methods of deformable solid mechanics and mathematical physics. In particular, the generalized Fourier method associated with the use of finite Fourier series and orthogonal splines was successfully applied earlier by the author in solving parabolic initial boundary value problems for regions with curved boundaries. Recent article proposes further development and novel full research of the algorithm of this method, designed to solve parabolic initial boundary value problems in non-canonical domains. The method gives approximate analytical solutions in form of finite Fourier series whose structure is similar to that of partial sums of an infinite Fourier series for an exact solution. Full investigation of the method’s convergence presented in this article is based on the theory of finite difference methods. As a number of grid nodes in a region increases, such finite Fourier series approach an exact solution of a parabolic initial boundary value problem. Investigation of convergence shows efficiency of the novel algorithm of the generalized Fourier method in solving parabolic initial boundary value problems for non-canonical regions.
Keywords: parabolic initial boundary value problems, curvilinear boundary, non-canonical regions, method of variable separation 
@article{SVMO_2024_26_3_a1,
     author = {V. L. Leontev},
     title = {Convergence of {Fourier} method connected with orthogonal splines},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {245--259},
     publisher = {mathdoc},
     volume = {26},
     number = {3},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2024_26_3_a1/}
}
TY  - JOUR
AU  - V. L. Leontev
TI  - Convergence of Fourier method connected with orthogonal splines
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2024
SP  - 245
EP  - 259
VL  - 26
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2024_26_3_a1/
LA  - ru
ID  - SVMO_2024_26_3_a1
ER  - 
%0 Journal Article
%A V. L. Leontev
%T Convergence of Fourier method connected with orthogonal splines
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2024
%P 245-259
%V 26
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2024_26_3_a1/
%G ru
%F SVMO_2024_26_3_a1
V. L. Leontev. Convergence of Fourier method connected with orthogonal splines. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 3, pp. 245-259. http://geodesic.mathdoc.fr/item/SVMO_2024_26_3_a1/

[1] V. L. Leontiev, “Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary”, Ufa Mathematical Journal, 14:2 (2022), 56–66 | DOI | MR | Zbl

[2] E. A. Gasymov, A. O. Guseinova, U. N. Gasanova, “Application of generalized separation of variables to solving mixed problems with irregular boundary conditions”, Computational Mathematics and Mathematical Physics, 56:7 (2016), 1305–1309 | DOI | MR | Zbl

[3] I. A. Savichev, A. D. Chernyshov, “Application of the angular superposition method to the contact problem on the compression of an elastic cylinder”, Mechanics of Solids, 44:3 (2009), 463–472 | DOI

[4] Yu. I. Malov, L. K. Martinson, K. B.Pavlov, “Solution by separation of the variables of some mixed boundary value problems in the hydrodynamics of conducting media”, USSR Computational Mathematics and Mathematical Physics, 12:3 (1972), 71–86 | DOI | MR

[5] M. Sh.Israilov, “Diffraction of Acoustic and Elastic Waves on a Half-Plane for Boundary Conditions of Various Types”, Mechanics of Solids, 48:3 (2013), 337–347 | DOI

[6] A. Y. Majeed, L. G. Juan, O. M. Pshtiwan, N. Chorfi, D. Baleanu, “A computational study of time-fractional gas dynamics models by means of conformable finite difference method”, AIMS Mathematics, 9:7 (2024), 19843–19858 | DOI | MR

[7] R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, Society for Industrial and Applied Mathematics, 2007, 356 pp. | MR | Zbl

[8] Finite Difference Methods. Theory and Applications. 7th International Conference, Lozenetz, Bulgaria, 2018, 701 pp. | DOI

[9] D. Kh. Ivanov, P. N. Vabishchevich, “Iterative Process for Numerical Recovering the Lowest Order Space-Wise Coefficient in Parabolic Equations”, Finite Difference Methods. Theory and Applications. 7th International Conference, Lozenetz, Bulgaria, 2018, 289–296 | DOI | MR

[10] G. Strang, G. J. Fix, An Analysis of the Finite Element Method, Englewood Cliffs, 1973, 349 pp. | MR | Zbl

[11] P. Li, “A Fourier series model and parametric study for single person three-way continuous walking load”, Transactions on Computer Science and Intelligent Systems Research, 4 (2024) | DOI

[12] H. N. Amadi, K. O. Uwho, D. M. Kornom, “Modelling of Electrical Cable Parameters and Fault Detection using Fourier Series”, Journal of Recent Trends in Electrical Power System, 7:2 (2024), 44–56 | DOI

[13] G. F. Voronin, S. A. Degtyarev, “Spline-approximation of thermodynamic properties of solutions”, Computer Coupling of Phase Diagrams and Thermochemistry, 6:3 (1982), 217–227 | DOI

[14] V. L. Leontiev, “Orthogonal Splines in Approximation of Functions”, Mathematics and Statistics, 8:2 (2020), 167–172 | DOI | MR

[15] V. L. Leontiev, Orthogonal splines and special functions in methods of computational mechanics and mathematics, POLITEKH-PRESS, St. Petersburg, 2021, 466 pp. (In Russ.)

[16] E. A. Volkov, Numerical Methods, Hemisphere Publishing Corp., New York, 1990, 238 pp. | MR

[17] S. K. Godunov, V. S.Ryabenkii, Difference Schemes. An Introduction to the Underlying Theory, Elsevier Science Publishers B.V., 1987., 489 pp. | DOI | MR