Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Engineering with Computers
Published in: Engineering with Computers 4/2020

13-06-2019 | Original Article

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

Authors: Sidra Saleem, Imran Aziz, Malik Zawwar Hussain

Published in: Engineering with Computers | Issue 4/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.
previous article New collocation method for stochastic response surface reliability analyses
next article Enhanced leadership-inspired grey wolf optimizer for global optimization problems

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Tadmor E (2012) A review of numerical methods for non-linear partial differential equations. Bull Am Math Soc 49:507–554CrossRef
2.
Polyanin A, Zaitsev V (2018) Handbook of nonlinear partial differential equations. CRC Press, CambridgeMATH
3.
Shi D, Wang J (2017) Unconditional super convergence analysis of conforming finite element for nonlinear parabolic equation. Appl Math Comput 294:216–226MathSciNetCrossRef
4.
Benes M, Kruis J (2018) Multi-time-step domain decomposition and coupling methods for nonlinear parabolic problems. Appl Math Comput 319:444–460MathSciNetCrossRef
5.
Mohanty RK, Kaur D (2016) High accuracy implicit variable mesh methods for numerical study of special types of fourth order non-linear parabolic equations. Appl Math Comput 273:678–696MathSciNetCrossRef
6.
Mohanty RK, Setia N (2015) A new high accuracy two-level implicit off-step discretization for the system of three space dimensional quasi-linear parabolic partial differential equations. Comput Math Appl 69:1096–1113MathSciNetCrossRef
7.
Mohanty RK (2007) An implicit high accuracy variable mesh scheme for 1-D non-linear singular parabolic partial differential equations. Appl Math Comput 186:219–229MathSciNetCrossRef
8.
Boglaev I (2016) Numerical solving nonlinear integro-parabolic equations by the monotone weighted average method. Appl Math Comput 274:152–162MathSciNetCrossRef
9.
Boglaev I (2016) Monotone iterative ADI method for solving coupled systems of nonlinear parabolic equations. Appl Numer Math 108:204–222MathSciNetCrossRef
10.
Feng X, He Y (2016) Convergence of the Crank-Nicolson/Newton scheme for nonlinear parabolic problem. Acta Math Sci 36:124–138MathSciNetCrossRef
11.
Tan WC, Hoang VH (2019) High dimensional finite element method for multiscale nonlinear monotone parabolic equations. J Comput Appl Math 345:471–500MathSciNetCrossRef
12.
Abdulla UG, Du J, Prinkey A, Ondracek C, Parimoo S (2018) Evolution of interfaces for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption. Math Comput Simul 153:59–82MathSciNetCrossRef
13.
Koleva MN, Vulkov LG (2013) Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance. Math Comput Model 57:2564–2575MathSciNetCrossRef
14.
Horton G, Vandewalle S, Worley P (1995) An algorithm with polylog parallel complexity for solving parabolic partial differential equations. SIAM J Sci Comput 16:531–541MathSciNetCrossRef
15.
Chen C, Hsiao C (1997) Haar wavelet method for solving lumped and distributed parameter systems. IEEE Proc Contr Theor Appl 144:87–94CrossRef
16.
Lepik Ü (2007) Numerical solution of evolution equations by the Haar wavelet method. Appl Math Comput 185:695–704MathSciNetCrossRef
17.
18.
Majak J, Shvartsman BS, Kirs M, Pohlak M, Herranen H (2015) Convergence theorem for the Haar wavelet based discretization method. Compos Struct 126:227–232CrossRef
19.
Majak J, Shvartsman BS, Karjust K, Mikola M, Haavajoe A, Pohlak M (2015) On the accuracy of the Haar wavelet discretization method. Compos Part B Eng 80:321–327CrossRef
20.
Aziz I, Siraj-ul-Islam M (2017) Asif, Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Comput Math Appl 73:2023–2034MathSciNetCrossRef
21.
Aziz I, Amin R (2016) Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl Math Model 40:10286–10299MathSciNetCrossRef
22.
Siraj-ul-Islam, Aziz I, Šarler B (2010) The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math Comput Model 52:1577–1590MathSciNetCrossRef
23.
Aziz I, Siraj-ul-Islam, Šarler B (2013) Wavelets collocation methods for the numerical solution of elliptic BV problems. Appl Math Model 37:676–694MathSciNetCrossRef
24.
Majak J, Pohlak M, Eerme M, Lepikult T (2009) Weak formulation based Haar wavelet method for solving differential equations. Appl Math Comput 211:488–494MathSciNetCrossRef
25.
26.
27.
Metadata
Title
A simple algorithm for numerical solution of nonlinear parabolic partial differential equations
Authors
Sidra Saleem
Imran Aziz
Malik Zawwar Hussain
Publication date
13-06-2019
Publisher
Springer London
Published in
Engineering with Computers / Issue 4/2020
Print ISSN: 0177-0667
Electronic ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-019-00796-z

Other articles of this Issue 4/2020

Engineering with Computers 4/2020 Go to the issue

Original Article

An efficient operator-splitting FEM-FCT algorithm for 3D chemotaxis models

Original Article

An upwind approximation combined with mixed volume element for a positive semi-definite contamination treatment from nuclear waste

Original Article

Generalizations of non-uniform rational B-splines via decoupling of the weights: theory, software and applications

Original Article

Structural shape optimization using Bézier triangles and a CAD-compatible boundary representation

Original Article

Nonlinear transient analysis of delaminated curved composite structure under blast/pulse load

Original Article

2D Mesh smoothing based on Markov chain method