Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation

Dalabaev Umurdin; Hasanova Dilfuza

doi:doi:10.11648/j.mcs.20230802.11

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Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation

Dalabaev Umurdin, Hasanova Dilfuza

Published in Mathematics and Computer Science (Volume 8, Issue 2)

Received: 11 February 2023 Accepted: 27 February 2023 Published: 9 March 2023

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Abstract

The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.

Published in	Mathematics and Computer Science (Volume 8, Issue 2)
DOI	10.11648/j.mcs.20230802.11
Page(s)	39-45
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Parabolic Equation, Approximate-Analytical Solution, Moving Nodes, Method of Line

References

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[2]	Aramanovich I. G., Levin V. I. Equations of mathematical physics. M.: publishing house "Nauka", 1964, p. 162.
[3]	Martinson L. K., Malov Yu. I. Differential Equations of Mathematical Physics. M.: Publishing house of MSTU im. N. E. Bauman, 2002. 368 p.
[4]	S. G. Mikhlin, Variational methods for solving problems of mathematical physics, Uspekhi Mat. Nauk, 1950, volume 5, issue 6, 3–51.
[5]	K. Rectoris VARIATIONAL METHODS IN MATHEMATICAL PHYSICS AND TECHNOLOGY, M.: Mir, 1985.— 590 p.
[6]	H.-J. Reinhardt Projection Methods for Variational Equations, Part of the Applied Mathematical Sciences book series (AMS, volume 57).
[7]	Tamer A. Abassya, Magdy A. El-Tawilb,_, H. El-Zoheiryb. Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Pad´e technique, Computers and Mathematics with Applications 54 (2007) 940–954.
[8]	Dalabaev U. Difference-Analytical Method Of The One-Dimensional Convection-Diffusion equation IJISET – International Journal of Innovative Science. Engineering & Technology. Vol. 3. Issue 1. 2016. January. ISSN 2348 – 7968. – С. 234-239.
[9]	Dalabaev U. Computing Technology of a Method of Control Volume for Obtaining of the Approximate Analytical Solution to One-Dimensional Convection-Diffusion Problems // Open Access Library Journal, 2018. https://doi.org/10.4236/oalib.1104962 (2018).
[10]	U. Dalabaev Moving Node Method, Monograph, Generis Publishing, ISBN: 978-1-63902-785-9, 70 pp., 2022.
[11]	O. A. Liskovets, The method of lines, Differ. equations, 1965, volume 1, number 12, 1662–1678.
[12]	I. K. Karimov, I. K. Khuzhaev, and Zh. I. Khuzhaev, “Application of the method of lines in solving a one-dimensional parabolic type equation under boundary conditions of the second and first kind,” Vestnik KRAUNC. Phys.-Math. sciences, 2018, number 1, 78–92.
[13]	Schiesser W. E., “The numerical methods of lines”, San Diego, CA: Academic Press, 1991.
[14]	Michael B. C. And H. W. Hinds. “The method of lines and the advactive Equation. Simulation”, 31: 59-69, 1978.
[15]	Schiesser W. E., “The Numerical Method of Lines: Integration of Partial Differential Equations”, Academic Press, San Diego, Calif., 1991. 326p.
[16]	Patel K. B., “A numerical solution of travelling wave that has an increasingly steep moving front using Method of Lines” International journal of education and mathematics, vol -2 Feb.- 13. pp. 01-07.

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APA Style

Dalabaev Umurdin, Hasanova Dilfuza. (2023). Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Mathematics and Computer Science, 8(2), 39-45. https://doi.org/10.11648/j.mcs.20230802.11

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ACS Style

Dalabaev Umurdin; Hasanova Dilfuza. Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Math. Comput. Sci. 2023, 8(2), 39-45. doi: 10.11648/j.mcs.20230802.11

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AMA Style

Dalabaev Umurdin, Hasanova Dilfuza. Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Math Comput Sci. 2023;8(2):39-45. doi: 10.11648/j.mcs.20230802.11

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@article{10.11648/j.mcs.20230802.11,
  author = {Dalabaev Umurdin and Hasanova Dilfuza},
  title = {Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation},
  journal = {Mathematics and Computer Science},
  volume = {8},
  number = {2},
  pages = {39-45},
  doi = {10.11648/j.mcs.20230802.11},
  url = {https://doi.org/10.11648/j.mcs.20230802.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.11},
  abstract = {The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.},
 year = {2023}
}

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TY - JOUR
T1 - Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation
AU - Dalabaev Umurdin
AU - Hasanova Dilfuza
Y1 - 2023/03/09
PY - 2023
N1 - https://doi.org/10.11648/j.mcs.20230802.11
DO - 10.11648/j.mcs.20230802.11
T2 - Mathematics and Computer Science
JF - Mathematics and Computer Science
JO - Mathematics and Computer Science
SP - 39
EP - 45
PB - Science Publishing Group
SN - 2575-6028
UR - https://doi.org/10.11648/j.mcs.20230802.11
AB - The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.
VL - 8
IS - 2
ER -

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Author Information

Dalabaev Umurdin

Department of System Analysis and Mathematical Modeling, University of World Economy and Diplomacy, Tashkent, Uzbekistan
Hasanova Dilfuza

Department of System Analysis and Mathematical Modeling, University of World Economy and Diplomacy, Tashkent, Uzbekistan

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APA Style

Dalabaev Umurdin, Hasanova Dilfuza. (2023). Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Mathematics and Computer Science, 8(2), 39-45. https://doi.org/10.11648/j.mcs.20230802.11

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ACS Style

Dalabaev Umurdin; Hasanova Dilfuza. Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Math. Comput. Sci. 2023, 8(2), 39-45. doi: 10.11648/j.mcs.20230802.11

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AMA Style

Dalabaev Umurdin, Hasanova Dilfuza. Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation. Math Comput Sci. 2023;8(2):39-45. doi: 10.11648/j.mcs.20230802.11

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@article{10.11648/j.mcs.20230802.11,
  author = {Dalabaev Umurdin and Hasanova Dilfuza},
  title = {Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation},
  journal = {Mathematics and Computer Science},
  volume = {8},
  number = {2},
  pages = {39-45},
  doi = {10.11648/j.mcs.20230802.11},
  url = {https://doi.org/10.11648/j.mcs.20230802.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.11},
  abstract = {The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.},
 year = {2023}
}

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