An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations

Okey Oseloka Onyejekwe

doi:doi:10.11648/j.acm.20150403.11

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An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations

Okey Oseloka Onyejekwe

Published in Applied and Computational Mathematics (Volume 4, Issue 3)

Received: 20 March 2015 Accepted: 29 March 2015 Published: 18 April 2015

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Abstract

This paper explores the application of an Hermitian hybrid boundary integral formulation for handling Fisher-type equations. The Hermite system incorporates the problem unknowns with their space derivatives and as a consequence produces a relatively larger coefficient matrix than the corresponding linear approximation. However by adopting a finite element-like integral numerical procedure, the modified boundary integral formulation otherwise known as the Green element method (GEM) produces slender and sparse coefficient matrices which enhance an efficient solution algorithm. The resulting equations appear in the form of local elemental integral equations whose contributions add up to the coefficient matrix. This process is amply simplified in the Green element method due to the presence of the source point inside an element thereby encouraging integration to be carried out locally and accurately. This so called ‘divide and conquer’ approach is significantly much better than working with the entire matrix especially for nonlinear problems where an encounter with the problem domain can not be totally avoided. Numerical tests are carried out to illustrate the utility of this technique by comparing results obtained from both the Hermite and non-Hermite discretizations. It is observed that for each of the problems tested, not only do the results agree with those from literature, it took the Hermitian approximation fewer number of elements to achieve the same level of accuracy than its non-Hermitian version. However, application of same technique to multi-dimensional problems may not be as straightforward due to the construction and storing of the Hermite system matrix which will not only involve non-trivial operations in terms of a high computational cost but also a compromise in the quality of the numerical solution arising from significant round-off errors.

Published in	Applied and Computational Mathematics (Volume 4, Issue 3)
DOI	10.11648/j.acm.20150403.11
Page(s)	83-99
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

Fisher-Type Differential Equations, Boundary Element Method, Nonlinearity, Hybrid Boundary Integral Method, Green Element Method, Hermite Interpolation

References

[1]	R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), pp.355-369
[2]	J. Canosa, On a nonlinear equation describing population growth, IBM J. Res. Develop 17 (1973), pp 307-313
[3]	N. Parekh, S. Puri, A new numerical scheme for the Fishers equation Journal of PhysicsA., 23 (1990) pp 1085-1091
[4]	M.S. El-Azab, An approximation scheme for a nonlinear diffusion Fisher’s equation, Appld. Math. Comput. 186 (2007) , pp 579-588
[5]	D. Olmos and B.D. Shizgal A pseudospectral method of solution of Fisher’s equation, J. Comput. Appld. Math. 193 (2007), pp. 721-735
[6]	W. Guo Cheng Uniformly constructing soliton solutions and solutions to Burgers Fisher equation, Computer and Mathematics with Applic. 56 (2009) 2355-2357
[7]	W. Graham Griffiths, and E. William Schiesser Burgers-Fisher equation travelling wave analysis of partial differential equations of partial differential analysis of partial differential equations, numerical and analytical methods with Matlab and Maple, Academic Press (2012) m pp 123-133
[8]	M. Javidi, Spectral collocation method for the generalized Burger-Fisher equation, Applied mathematics and computation 216 (9) (2010) pp 2679-2686
[9]	D. Olmos and B. D. Shizgal, A pseudospectral method of solution of Fishers equation, Journal of Computational and Applied Mathematics 193 (2006) pp219-242
[10]	J. Gazdag and J. Canosa, Numerical solution of Fisher’s equation J. Applied Probab. 11 (1974), pp. 445-457
[11]	R.C. Mittal and S. Kumar, Numerical study of Fisher’s equation by wavelet Galerkins method Int. Jnl. Comput. Math 83(2006) 287-298
[12]	R.E. Mickens A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion , Computers and Mathematics with Application 45 (1-3) (2003) pp 429-436
[13]	R. Uddin Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation, SIAM Jnl. Sci. Comput. 22 (6) (2006) pp 1926-1942
[14]	C.A.B. Chaverra, H. Power, W.F. Florez Escobar, A.F.H. Betancourt Two-dimensional meshless solution of the nonlinear convection-diffusion-reaction by ht elocal Hermitian interpolation method Ingeniera Y Ciencia 9(17) (2013) pp. 21-51
[15]	V. Popov and H. Power The DRM-MD Integral equation method : An efficient approach for the numerical solution of domain dominant problems, Int. J. Num. Methods in Engineering 44 (1999) pp 327-353
[16]	T. Mavoungou, Y. Cherruault, Numerical study of Fisher’s equation by Adomain’s method, Mathematical and Computer Modelling 19 (1) (1994) pp 89-95
[17]	A. Sahin, I. Dag, B Saka A B-spline algorithm for the numerical solution Fisher’s equation, Kybernetes 37 (2) (2008) pp. 326-342
[18]	G. Meral and M. Tezer-Sezgin Solution of nonlinear reaction-diffusion equation by using dual reciprocity boundary element method with finite difference or least square method, Advances in Boundary Element Techniques IX Eds. R. Abascal, N.H. Aliabadi 2008
[19]	A.E. Taigbenu The Green element method Int. Journal. Num. Mthds. Engnr. 38 (1995) pp. 2241-2263
[20]	A.E. Taigbenu The Green element method Kluwer Acad. Publishers, Boston 376 pp
[21]	A.E. Taigbenu, O.O. Onyejekwe Green’s function –based integral approaches to nonlinear transient boundary-value problems (II) Appld. Math Model. 23 (1999) pp 241-253
[22]	J.C. Wu Fundamental solutions and numerical methods for flow problems Int. Jnl. Num. Mthds Fluids 4 (1984) pp 185-201
[23]	J. Sladeck, V. Sladeck, C. Zhang A local BIEM for analysis of transient heat conduction with nonlinear source terms in FGMs, Engnr. Analysis Boundary Elem. 28 (2004) pp 1-11
[24]	O.O. Onyejekwe The effect of time stepping schemes on the accuracy of Green element formulation of unsteady transport Jnl. Appld. Math. And Physics 2 (2014) pp 621-633
[25]	M. Hibersek, and L. Skerget Domain decomposition methods for fluid flow problems by the boundary integral method Zeitschrift fur Angewandle Mathematik und Mechanik 76, (1996) pp 115-139
[26]	M. Portapila and H. Power Iterative solution schemes for quadratic DRM-MD Numerical Mthds. Partial Diff. Equatn. 24 (2008) pp1430-1459
[27]	J. Gray and M. Garzon, On a Hermite boundary integral approximation Computers and structures, 83 (2005) pp. 889-894
[28]	R. Archer Continuous solutions from the Green element method using Overhauser elements, Applied Numerical Math. 56 (2) (2006) pp 222-229
[29]	A.E. Taigbenu, Numerical stability characteristics of a hermitian Green element model for transport equations, Engineering Analysis with Boundary Elements 22 (2) (1998) pp. 161-165
[30]	O.O. Onyejekwe Experiences in solutions of non-linear transport equations with Green element method, Int. Jnl. Numer. Mthds Heat and Fluid Flow, 7 (2000) pp 675-686
[31]	O. O. Onyejekwe and O. N. Onyejekwe A nonlinear integral formulation for a steam-aquifer interaction flow problem, International Jnl. Nonlinear Science, 15 (1) (2013) pp15-26
[32]	M.J. Ablowitz and A. Zepetella, Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math. Biol. 41 (1979) pp. 835-840
[33]	J.I. Ramos, A review of some numerical methods for reaction-diffusion equations, Mathematics and Computers in Simulation XXV (1983) pp. 538-548
[34]	R.C. Mittal and R.K. Jain Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B-spline collocation method, Math. Sci 7 (12) (2013) pp. 1-10
[35]	C. Cattani and A. Kudreyko A multiscale analysis of the Fisher equation. In: (ed) ICCSA proceedings of the international conference on computational science and its applications, part 1, lecture notes in computer science 5072nd edn, pp 117- 1180. Springer Berlin (2008)
[36]	R. Lin and H. Zhu A discontinuous Galerkin least-squares finite element method for solving Fisher’s equation, Discrete and Continuous Dynamical Systems (2013) pp. 489-497
[37]	S. Li, L.Petzold and Y. Ren, Stabikity of moving mesh systems of partial differential ,SIAM L, Sci, 20 (1998) pp. 719-738
[38]	Y. Qiu and D. M. Sloan Numerical solution of Fisher equation using a moving mesh method, J. Comput. Physics 146 (1998) pp 726-746
[39]	A.Segal, N. Praagman,A fast implementation of of eplicit time-stepping algorithms with finite element method for a class of nonlinear evolution problems, Int. Jnl. Num. Mthds. Engnr. 23 (1986) pp. 155- 168

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Okey Oseloka Onyejekwe. (2015). An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Applied and Computational Mathematics, 4(3), 83-99. https://doi.org/10.11648/j.acm.20150403.11

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Okey Oseloka Onyejekwe. An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Appl. Comput. Math. 2015, 4(3), 83-99. doi: 10.11648/j.acm.20150403.11

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AMA Style

Okey Oseloka Onyejekwe. An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Appl Comput Math. 2015;4(3):83-99. doi: 10.11648/j.acm.20150403.11

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@article{10.11648/j.acm.20150403.11,
  author = {Okey Oseloka Onyejekwe},
  title = {An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {83-99},
  doi = {10.11648/j.acm.20150403.11},
  url = {https://doi.org/10.11648/j.acm.20150403.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.11},
  abstract = {This paper explores the application of an Hermitian hybrid boundary integral formulation for handling Fisher-type equations. The Hermite system incorporates the problem unknowns with their space derivatives and as a consequence produces a relatively larger coefficient matrix than the corresponding linear approximation. However by adopting a finite element-like integral numerical procedure, the modified boundary integral formulation otherwise known as the Green element method (GEM) produces slender and sparse coefficient matrices which enhance an efficient solution algorithm. The resulting equations appear in the form of local elemental integral equations whose contributions add up to the coefficient matrix. This process is amply simplified in the Green element method due to the presence of the source point inside an element thereby encouraging integration to be carried out locally and accurately. This so called ‘divide and conquer’ approach is significantly much better than working with the entire matrix especially for nonlinear problems where an encounter with the problem domain can not be totally avoided. Numerical tests are carried out to illustrate the utility of this technique by comparing results obtained from both the Hermite and non-Hermite discretizations. It is observed that for each of the problems tested, not only do the results agree with those from literature, it took the Hermitian approximation fewer number of elements to achieve the same level of accuracy than its non-Hermitian version. However, application of same technique to multi-dimensional problems may not be as straightforward due to the construction and storing of the Hermite system matrix which will not only involve non-trivial operations in terms of a high computational cost but also a compromise in the quality of the numerical solution arising from significant round-off errors.},
 year = {2015}
}

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TY  - JOUR
T1  - An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations
AU  - Okey Oseloka Onyejekwe
Y1  - 2015/04/18
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.11
DO  - 10.11648/j.acm.20150403.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 83
EP  - 99
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150403.11
AB  - This paper explores the application of an Hermitian hybrid boundary integral formulation for handling Fisher-type equations. The Hermite system incorporates the problem unknowns with their space derivatives and as a consequence produces a relatively larger coefficient matrix than the corresponding linear approximation. However by adopting a finite element-like integral numerical procedure, the modified boundary integral formulation otherwise known as the Green element method (GEM) produces slender and sparse coefficient matrices which enhance an efficient solution algorithm. The resulting equations appear in the form of local elemental integral equations whose contributions add up to the coefficient matrix. This process is amply simplified in the Green element method due to the presence of the source point inside an element thereby encouraging integration to be carried out locally and accurately. This so called ‘divide and conquer’ approach is significantly much better than working with the entire matrix especially for nonlinear problems where an encounter with the problem domain can not be totally avoided. Numerical tests are carried out to illustrate the utility of this technique by comparing results obtained from both the Hermite and non-Hermite discretizations. It is observed that for each of the problems tested, not only do the results agree with those from literature, it took the Hermitian approximation fewer number of elements to achieve the same level of accuracy than its non-Hermitian version. However, application of same technique to multi-dimensional problems may not be as straightforward due to the construction and storing of the Hermite system matrix which will not only involve non-trivial operations in terms of a high computational cost but also a compromise in the quality of the numerical solution arising from significant round-off errors.
VL  - 4
IS  - 3
ER  -

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Author Information

Okey Oseloka Onyejekwe

Computational Science Program, Addis Ababa University, Arat Kilo campus, Addis Ababa, Ethiopia

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Okey Oseloka Onyejekwe. (2015). An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Applied and Computational Mathematics, 4(3), 83-99. https://doi.org/10.11648/j.acm.20150403.11

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ACS Style

Okey Oseloka Onyejekwe. An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Appl. Comput. Math. 2015, 4(3), 83-99. doi: 10.11648/j.acm.20150403.11

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Okey Oseloka Onyejekwe. An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations. Appl Comput Math. 2015;4(3):83-99. doi: 10.11648/j.acm.20150403.11

Copy | Download

@article{10.11648/j.acm.20150403.11,
  author = {Okey Oseloka Onyejekwe},
  title = {An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {83-99},
  doi = {10.11648/j.acm.20150403.11},
  url = {https://doi.org/10.11648/j.acm.20150403.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.11},
  abstract = {This paper explores the application of an Hermitian hybrid boundary integral formulation for handling Fisher-type equations. The Hermite system incorporates the problem unknowns with their space derivatives and as a consequence produces a relatively larger coefficient matrix than the corresponding linear approximation. However by adopting a finite element-like integral numerical procedure, the modified boundary integral formulation otherwise known as the Green element method (GEM) produces slender and sparse coefficient matrices which enhance an efficient solution algorithm. The resulting equations appear in the form of local elemental integral equations whose contributions add up to the coefficient matrix. This process is amply simplified in the Green element method due to the presence of the source point inside an element thereby encouraging integration to be carried out locally and accurately. This so called ‘divide and conquer’ approach is significantly much better than working with the entire matrix especially for nonlinear problems where an encounter with the problem domain can not be totally avoided. Numerical tests are carried out to illustrate the utility of this technique by comparing results obtained from both the Hermite and non-Hermite discretizations. It is observed that for each of the problems tested, not only do the results agree with those from literature, it took the Hermitian approximation fewer number of elements to achieve the same level of accuracy than its non-Hermitian version. However, application of same technique to multi-dimensional problems may not be as straightforward due to the construction and storing of the Hermite system matrix which will not only involve non-trivial operations in terms of a high computational cost but also a compromise in the quality of the numerical solution arising from significant round-off errors.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - An Hermitian Boundary Integral Hybrid Formulation for Nonlinear Fisher-Type Equations
AU  - Okey Oseloka Onyejekwe
Y1  - 2015/04/18
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.11
DO  - 10.11648/j.acm.20150403.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 83
EP  - 99
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150403.11
AB  - This paper explores the application of an Hermitian hybrid boundary integral formulation for handling Fisher-type equations. The Hermite system incorporates the problem unknowns with their space derivatives and as a consequence produces a relatively larger coefficient matrix than the corresponding linear approximation. However by adopting a finite element-like integral numerical procedure, the modified boundary integral formulation otherwise known as the Green element method (GEM) produces slender and sparse coefficient matrices which enhance an efficient solution algorithm. The resulting equations appear in the form of local elemental integral equations whose contributions add up to the coefficient matrix. This process is amply simplified in the Green element method due to the presence of the source point inside an element thereby encouraging integration to be carried out locally and accurately. This so called ‘divide and conquer’ approach is significantly much better than working with the entire matrix especially for nonlinear problems where an encounter with the problem domain can not be totally avoided. Numerical tests are carried out to illustrate the utility of this technique by comparing results obtained from both the Hermite and non-Hermite discretizations. It is observed that for each of the problems tested, not only do the results agree with those from literature, it took the Hermitian approximation fewer number of elements to achieve the same level of accuracy than its non-Hermitian version. However, application of same technique to multi-dimensional problems may not be as straightforward due to the construction and storing of the Hermite system matrix which will not only involve non-trivial operations in terms of a high computational cost but also a compromise in the quality of the numerical solution arising from significant round-off errors.
VL  - 4
IS  - 3
ER  -

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