﻿ Development of a Numerical Scheme

American Journal of Computational Mathematics
Vol.06 No.01(2016), Article ID:65182,6 pages
10.4236/ajcm.2016.61006

Development of a Numerical Scheme

R. B. Ogunrinde, T. E. Olaosebikan

Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

Copyright © 2016 by authors and Scientific Research Publishing Inc.

Received 19 February 2016; accepted 27 March 2016; published 30 March 2016

ABSTRACT

In this paper, we developed a new numerical scheme which aimed to solve some initial value problems of ordinary differential equations. The full breakdown of this new numerical scheme derivation is presented. While in our subsequent research, we shall fully examine the characteristics of the scheme such as consistency, convergence and stability. Also, the implementation of this new numerical scheme shall be worked-on and comparison shall also be made with some existing methods.

Keywords:

Numerical Scheme, Ordinary Differential Equation, Scheme Development

1. Introduction

Many numerical analysts such as: S. O. Fatunla [1] , E. A. Ibijola [2] [3] , R. B. Ogunrinde [4] and even A. A. Obayomi [5] and so on, have developed schemes for the solution of some initial value problem of ordinary differential equations. The efficiency of all these contributed effort from this numerical analyst in numerical analysis had been measured and tested for their stability, accuracy, convergence and consistency properties. The accuracy properties of different methods are usually compared by considering the order of convergence as well as the truncation error coefficients of the various methods (C. F. Tischer, 1984). From literatures, this shows that so many methods which are suitable for solving some sets of initial value problems (ivps) in ordinary differential equations (ODEs) must have all the mentioned characteristics.

Ogunrinde, R. B. [4] , developed a scheme in which standard finite difference schemes were developed. Similarly, Obayomi, A. A. [5] [6] , also worked on some approximation techniques which was used to derive qualitatively stable non-standard finite difference schemes.

In this paper, a new numerical scheme was developed with the above mentioned characteristics in mind to solve some initial value problems of ordinary differential equations which was based on the local representation of the theoretical solution to initial value problem of the form:

in the interval by interpolating function , where, , and b are real undetermined coefficients.

2. Derivation of the New Scheme

Suppose we have the initial value problem:

(1)

Let us assume that the theoretical solution to (1) can be locally represented in the interval, by the interpolating polynomial function:

(2)

where, , , and b are real undetermined coefficients.

We shall assume that is a numerical estimate to the theoretical solution and. We define mesh points as follows:

Therefore, from (2), we proceed to the scheme derivation as follows:

(3)

(4)

(5)

(6)

from (2),

(7)

from (3),

(8)

from (4),

(9)

from (5),

(10)

putting (8) into (9), we have:

multiply through by, we have:

(11)

putting (11) into (10), we obtain:

(12)

putting (12) into (11), we obtained:

(13)

putting (12) and (13) into (8), we have:

Now,

(14)

Let

Therefore,

(14a)

Now, imposing the following constraints on the interpolating function (2) in the following order:

1) The interpolating function (2) must coincide with the theoretical solution at and such that:

2) The derivative of and coincide with and respectively. i.e.

from conditions (1) and (2) above, it follows that:

if, then, we have:

Collecting like-terms

So,

(15)

Now, suppose:

(16)

Also,

(17)

from (15), we have:

(18)

Similarly,

by factorization, we have:

(19)

(20)

Putting (16) through (20) into (15), we have the new scheme follows:

(21)

Equation (21) is the proposed scheme.

3. Conclusions

We aim to develop a new numerical scheme which can favourably agree with the existing ones for solving some initial value problems of ordinary differential equations. Clearly, this paper has been able to show the development of the new numerical scheme as proposed.

In our subsequent research, we shall pay more attention on the implementation of this new scheme to solve some initial value problems (ivp) of the form (1) and also compare the results with the existing methods and thereafter we examine the characteristics properties such as the stability, convergence, accuracy and consistency of the scheme.

Cite this paper

Yu-Wen Chen,Der-Shing Lee,R. B. Ogunrinde,T. E. Olaosebikan, (2016) Development of a Numerical Scheme. American Journal of Computational Mathematics,06,49-54. doi: 10.4236/ajcm.2016.61006

References

1. 1. Bischoff, B.L. and Anderson, M.A. (1995) Peptization Process in the Sol-Gel Preparation of Porous Anatase TiO2. Chemical Materials, 7, 1772-1778.
http://dx.doi.org/10.1021/cm00058a004

2. 2. Chrysicopoulou, P., Davazoglou, D., Trapalis, C. and Kordas, G. (1998) Optical Properties of Very Thin (<100 nm) Sol-Gel TiO2 Films. Thin Solid Films, 323, 188-193.
http://dx.doi.org/10.1016/S0040-6090(97)01018-3

3. 3. Dibble, L.A. and Raupp, G.B. (1992) Fluidized-Bed Photocatalytic Oxidation of Trichloroethylene in Contaminated Airstreams. Environmental Science and Technology, 26, 492-495.
http://dx.doi.org/10.1021/es00027a006

4. 4. Cojocaru, B., Neatu, S., Parvulescu, V.I., Somoghi, V., Petrea, N., Epure, G., Alvaro, M. and Garcia, H. (2009) Synergism of Activated Carbon and Undoped and Nitrogen-Doped TiO2 in the Photocatalytic Degradation of the Chemical Warfare Agents. ChemSusChem, 2, 427-436.
http://dx.doi.org/10.1002/cssc.200800246

5. 5. Arabatzis, I.M., Stergiopoulos, T., Bernard, M.C., Labou, D., Neophytides, S.G. and Falaras, P. (2003) Silver-Modified Titanium Dioxide Thin Films for Efficient Photodegradation of Methyl Orange. Applied Catalysis B: Environmental, 42, 187-201.
http://dx.doi.org/10.1016/S0926-3373(02)00233-3

6. 6. Aruna, S.T., Tirosh, S. and Zaban, A. (2000) Nanosize Rutile Titania Particle Synthesis via a Hydrothermal Method without Mineralizers. Journal of Materials Chemistry, 10, 2388-2391.
http://dx.doi.org/10.1039/b001718n

7. 7. Asahi, R., Morikawa, T., Ohwaki, T., Aoki, K. and Taga, Y. (2001) Visible-Light Photocatalysis in Nitrogen-Doped Titanium Oxides. Science, 293, 269-271.
http://dx.doi.org/10.1126/science.1061051

8. 8. Babapour, A., Akhavan, O., Azimirad, R. and Moshfegh, A.Z. (2006) Physical Characteristics of Heat-Treated Nano-Silvers Dispersed in Sol-Gel Silica Matrix. Nanotechnology, 17, 763-771.
http://dx.doi.org/10.1088/0957-4484/17/3/025

9. 9. Balek, V., Li, D., Subrt, J., Vecerníková, E., Hishita, S., Mitsuhashi, T. and Haneda, H. (2007) Characterization of Nitrogen and Fluorine Co-Doped Titaniaphotocatalyst: Effect of Temperature on Microstructure and Surface Activity Properties. Journal of Physical Chemistry and Solids, 68, 770-774.
http://dx.doi.org/10.1016/j.jpcs.2007.01.028

10. 10. Carp, O., Huisman, C.L. and Reller, A. (2004) Photoinduced Reactivity of Titanium Dioxide. Progress in Solid State Chemistry, 32, 33-177.
http://dx.doi.org/10.1016/j.progsolidstchem.2004.08.001

11. 11. Li, C.H., Hsieh, Y.H., Chiu, W.T., Liu, C.C. and Kao, C.L. (2007) Study on Preparation and Photocatalytic Performance of Ag/TiO2 and Pt/TiO2 Photocatalysts. Separation and Purification Technology, 58, 148-151.
http://dx.doi.org/10.1016/j.seppur.2007.07.013

12. 12. Li, F.B. and Li, X.Z. (2002) Photocatalytic Properties of Gold/Gold Ion-Modified Titanium Dioxide for Wastewater Treatment. Applied Catalysis A: General, 228, 15-27.
http://dx.doi.org/10.1016/S0926-860X(01)00953-X

13. 13. Li, F.B. and Li, X.Z. (2002) The Enhancement of Photodegradation Efficiency Using Pt/TiO2 Catalyst. Chemosphere, 48, 1103-1111.
http://dx.doi.org/10.1016/S0045-6535(02)00201-1

14. 14. Li, J., Xu, J., Dai, W.L., Li, H. and Fan, K. (2009) Direct Hydro-Alcohol Thermal Synthesis of Special Core-Shell Structured Fe-Doped Titania Microspheres with Extended Visible Light Response and Enhanced Photoactivity. Applied Catalysis B: Environmental, 85, 162-170.
http://dx.doi.org/10.1016/j.apcatb.2008.07.008

15. 15. Haruta, M. (2007) Size- and Support-Dependency in the Catalysis of Gold. Catalysis Today, 36, 153-166.
http://dx.doi.org/10.1016/S0920-5861(96)00208-8

16. 16. Hashimoto, K., Irie, H. and Fujishima, A. (2005) TiO2 Photocatalysis: A Historical Overview and Future Prospects. Japanese Journal of Applied Physics, 44, 8269-8278.
http://dx.doi.org/10.1143/JJAP.44.8269

17. 17. Hirakawa, T. and Kamat, P.V. (2005) Charge Separation and Catalytic Activity of Ag@TiO2 Core-Shell Composite Clusters under UV-Irradiation. Journal of American Chemical Society, 127, 3928-3934.
http://dx.doi.org/10.1021/ja042925a

18. 18. Hirakawa, T. and Kanat, P.V. (2004) Photoinduced Electron Storage and Surface Plasmon Modulation in Ag@TiO2 Clusters. Langmuir, 20, 5645-5647.
http://dx.doi.org/10.1021/la048874c

19. 19. Hu, C., Hao, Z., Wong, P.K. and Yu, J.C. (2003) Photocatalytic Degradation of Triazine-Containing Azo Dyes in Aqueous TiO2 Suspensions. Applied Catalysis B: Environmental, 42, 47-55.
http://dx.doi.org/10.1016/S0926-3373(02)00214-X

20. 20. Yu, J.C., Yu, J., Yip, H., Wong, P.K., Zhao, J. and Ho, W. (2005) Efficient Visible-Light-Induced Photocatalytic Disinfection on Sulfur-Doped Nanocrystalline Titania. Environmental Science and Technology, 39, 1175-1179.
http://dx.doi.org/10.1021/es035374h

21. 21. Zhang, R. and Gao, L. (2002) Preparation of Nanosized Titania by Hydrolysis of Alkoxide Titanium in Micelles. Materials Research Bulletin, 37, 1659-1666.
http://dx.doi.org/10.1016/S0025-5408(02)00817-6

22. 22. Behar, D. and Rabani, J. (2006) Kinetics of Hydrogen Production upon Reduction of Aqueous TiO2 Nanoparticles Catalyzed by Pd0, Pt0, or Au0 Coatings and an Unusual Hydrogen Abstraction; Steady State and Pulse Radiolysis Study. Journal of Physical Chemistry B, 110, 8750-8755.
http://dx.doi.org/10.1021/jp060971m

23. 23. Kim, C.S., Moon, B.K., Park, J.H. and Son, S.M. (2003) Synthesis of Nanocrystalline TiO2 in Toluene by a Solvothermal Route. Journal of Crystal Growth, 254, 405-410.
http://dx.doi.org/10.1016/S0022-0248(03)01185-0

24. 24. Kim, S.B. and Hong, S.C. (2002) Kinetic Study for Photocatalytic Degradation of Volatile Organic Compounds in Air Using Thin Film TiO2 Photocatalyst. Applied Catalysis B: Environmental, 35, 305-315.
http://dx.doi.org/10.1016/S0926-3373(01)00274-0

25. 25. Kim, S.K., Choi, W. and Hwang, S.J. (2005) Visible Light Active Platinum-Ion-Doped TiO2 Photocatalyst. Journal of Physical Chemistry B, 109, 24260-24267.
http://dx.doi.org/10.1021/jp055278y

26. 26. Pastoriza-Santos, I., Mamedov, A.A., Giersig, M., Kotov, N.A., Liz-Marzán, L.M. and Koktysh, D.S. (2000) One-Pot Synthesis of Ag@TiO2 Core-Shell Nanoparticles and Their Layer-by-Layer Assembly. Langmuir, 16, 2731-2735.
http://dx.doi.org/10.1021/la991212g

27. 27. Chan, S.C. and Barteau, M.A. (2005) Preparation of Highly Uniform Ag/TiO2 and Au/TiO2 Supported Nanoparticle Catalysts by Photodeposition. Langmuir, 21, 5588-5595.
http://dx.doi.org/10.1021/la046887k

28. 28. Wang, W., Zhang, J., Chen, F., He, D. and Anpo, M. (2008) Preparation and Photocatalytic Properties of Fe3+-Doped Ag@TiO2 Core-Shell Nanoparticles. Journal of Colloid and Interfacial Science, 323, 182-186.
http://dx.doi.org/10.1016/j.jcis.2008.03.043

29. 29. Chuang, H.Y. and Chen, H. (2009) Fabrication and Photocatalytic Activities in Visible and UV Light Regions of Ag@TiO2 and NiAg@TiO2 Nanoparticles. Nanotechnology, 20, Article ID: 105704.
http://dx.doi.org/10.1088/0957-4484/20/10/105704

30. 30. Jia, H., Xu, H., Hu, Y., Tang, Y. and Zhang, L. (2007) TiO2@CdS Core-Shell Nanorods Films: Fabrication and Ramatically Enhanced Photoelectrochemical Properties. Electrochemical Communications, 9, 354-360.
http://dx.doi.org/10.1016/j.elecom.2006.10.010

31. 31. Li, X.Y., Yue, P.L. and Kutal, C. (2003) Synthesis and Photocatalytic Oxidation Properties of Iron Doped Titanium Dioxide Nanosemiconductor Particles. New Journal of Chemistry, 27, 1264-1269.
http://dx.doi.org/10.1039/b301998e

32. 32. Sakai, H., Kanda, T., Shibata, H., Ohkubo, T. and Abe, M. (2006) Preparation of Highly Dispersed Core/Shell-Type Titania Nanocapsules Containing a Single Ag Nanoparticle. Journal of American Chemical Society, 128, 4944-4945.
http://dx.doi.org/10.1021/ja058083c

33. 33. Tom, R.T., Nair, A.S., Singh, N., Aslam, N., Nagendra, C.L., Philip, R., Vijayamohanan, K. and Pradeep, T. (2003) Freely Dispersible Au@TiO2, Au@ZrO2, Ag@TiO2, and Ag@ZrO2 Core-Shell Nanoparticles: One-Step Synthesis, Characterization, Spectroscopy, and Optical Limiting Properties. Langmuir, 19, 3439-3445.
http://dx.doi.org/10.1021/la0266435

34. 34. Galindo, C., Jacques, P. and Kalt, A. (2001) Photooxidation of the Phenylazonaphthol AO20 on TiO2: Kinetic and Mechanistic Investigations. Chemosphere, 45, 997-1005.
http://dx.doi.org/10.1016/S0045-6535(01)00118-7

35. 35. Rauf, M.A. and Ashraf, S.S. (2009) Fundamental Principles and Application of Heterogeneous Photocatalytic Degradation of Dyes in Solution. Chemical Engineering Journal, 151, 10-18.
http://dx.doi.org/10.1016/j.cej.2009.02.026

36. 36. Lin, Y.C. and Lin, C.H. (2008) Catalytic and Photocatalytic Degradation of Ozone via Utilization of Controllable Nano-Ag Modified on TiO2. Environmental Progress, 27, 496-502.
http://dx.doi.org/10.1002/ep.10305

37. 37. Moulder, J.F., Stickle, W.F., Sobol, P.E. and Bomben, K.E. (1995) Handbook of X-Ray Photoelectron Spectroscopy. Physical Electronics.

38. 38. Li, X.Z., Li, F.B., Yang, C.L. and Ge, W.K. (2001) Photocatalytic Activity of WOx-TiO2 under Visible Light Irradiation. Journal of Photochemistry and Photobiology A: Chemistry, 141, 209-217.
http://dx.doi.org/10.1016/S1010-6030(01)00446-4

39. 39. Linsebigler, A.L., Lu, G. and Yates Jr., J.T. (1995) Photocatalysis on TiO2 Surfaces: Principles, Mechanisms, and Selected Results. Chemical Review, 95, 735-758.
http://dx.doi.org/10.1021/cr00035a013

40. 40. Martin, S.T., Morrison, C.L. and Hoffmann, M.R. (1994) Photochemical Mechanism of Size-Quantized Vanadium-Doped TiO2 Particles. Journal of Physical Chemistry, 98, 13695-13704.
http://dx.doi.org/10.1021/j100102a041

41. 41. Meichtry, J.M., Rivera, V., Iorio, Y.D., Rodríguez, H.B., Román, E.S., Grela, M. and Litter, M.I. (2009) Photoreduction of Cr(VI) Using Hydroxoaluminiumtricarboxymonoamide Phthalocyanine Adsorbed on TiO2. Photochemistry and Photobiological Science, 8, 604-612.

42. 42. Natarajan, C. and Nogami, G. (1996) Cathodicel Ectrodeposition of Nanocrystalline Titanium Dioxide Thin Films. Journal of Electrochemical Society, 143, 1547-1550.
http://dx.doi.org/10.1149/1.1836677

43. 43. O’Regan, B. and Gr&aumltzel, M. (1991) A Low-Cost, High-Efficiency Solar Cell Based on Dye-Sensitized Colloidal TiO2 Films. Nature, 353, 737-740.
http://dx.doi.org/10.1038/353737a0

44. 44. Ohno, T., Akiyoshi, M., Umebayashi, T., Asai, K., Mitsui, T. and Matsumura, M. (2004) Preparation of S-Doped TiO2 Photocatalysts and Their Photocatalytic Activities under Visible Light. Applied Catalysis A: General, 265, 115-121.
http://dx.doi.org/10.1016/j.apcata.2004.01.007

45. 45. Park, H.K., Moon, Y.T., Kim, D.K. and Kim, C.H. (1996) Formation of Monodisperse Spherical TiO2 Powders by Thermal Hydrolysis of Ti(SO4)2. Journal of American Ceramic Society, 79, 2727-2732.
http://dx.doi.org/10.1111/j.1151-2916.1996.tb09038.x

46. 46. Poznyak, S.K., Kokorin, A.I. and Kulak, A.I. (1998) Effect of Electron and Hole Acceptors on the Photoelectrochemical Behaviour of Nanocrystalline Microporous TiO2 Electrodes. Journal of Electroanalytic Chemistry, 442, 99-105.

47. 47. Pruden, A.L. and Ollis, D.F. (1983) Photoassisted Heterogeneous Catalysis: The Degradation of Trichloroethylene in Water. Journal of Catalysis, 82, 404-417.
http://dx.doi.org/10.1016/0021-9517(83)90207-5

48. 48. Sato, S. (1986) Photocatalytic Activity of NOx-Doped TiO2 in the Visible Light Region. Chemical Physics Letter, 123, 126-128.
http://dx.doi.org/10.1016/0009-2614(86)87026-9

49. 49. Sonawane, R.S., Kale, B.B. and Dongare, M.K. (2004) Preparation and Photo-Catalytic Activity of Fe-TiO2 Thin Films Prepared by Sol-Gel Clip Coating. Materials Chemistry and Physics, 85, 52-57.
http://dx.doi.org/10.1016/j.matchemphys.2003.12.007

50. 50. Sonawane, R.S., Hegde, H.G. and Dongare, M.K. (2003) Preparation of Titanium(IV) Oxide Thin-Film Photocatalyst by Sol-Gel Dip Coating. Materials Chemistry and Physics, 77, 744-750.
http://dx.doi.org/10.1016/S0254-0584(02)00138-4

51. 51. Vamathevan, V., Tse, H., Amal, R., Low, G. and McEvoy, S. (2001) Effects of Fe3+ and Ag+ Ions on the Photocatalytic Degradation of Sucrose in Water. Catalysis Today, 68, 201-208.
http://dx.doi.org/10.1016/S0920-5861(01)00301-7

52. 52. Wang, C.Y., Liu, C.Y., Zheng, X., Chen, J. and Shen, T. (1998) The Surface Chemistry of Hybrid Nanometer-Sized Particles I. Photochemical Deposition of Gold on Ultrafine TiO2 Particles. Colloids Surfaces A: Physicochemical and Engineering Aspects, 131, 271-280.
http://dx.doi.org/10.1016/S0927-7757(97)00086-1

53. 53. Wang, J., Zhao, H., Liu, X., Li, X., Xu, P. and Han, X. (2009) Formation of Ag Nanoparticles on Water-Soluble Anatase TiO2 Clusters and the Activation of Photocatalysis. Catalysis Communications, 10, 1052-1056.
http://dx.doi.org/10.1016/j.catcom.2008.12.060

54. 54. Xu, N., Shi, Z., Fan, Y., Dong, J., Shi, J. and Hu, M.Z.C. (1999) Effects of Particle Size of TiO2 on Photocatalytic Degradation of Methylene Blue in Aqueous Suspensions. Industrial Engineering Chemistry Research, 38, 373-379.
http://dx.doi.org/10.1021/ie980378u

55. 55. Yildiz, A., Lisesivdin, S.B., Kasap, M. and Mardare, D. (2009) Non-Adiabatic Small Polaron Hopping Conduction in Nb-Doped TiO2 Thin Film. Physica B: Condensed Matter, 404, 1423-1426.
http://dx.doi.org/10.1016/j.physb.2008.12.034

56. 56. Yin, S., Fujishiro, Y., Wu, J., Aki, M. and Sato, T. (2003) Synthesis and Photocatalytic Properties of Fibrous Titania by Solvothermal Reactions. Journal of Materials Processing Technology, 137, 45-48.
http://dx.doi.org/10.1016/j.physb.2008.12.034

57. 57. Zhang, F., Pi, Y., Cui, J., Zhang, X., Guan, N. and Yang, Y. (2007) Unexpected Selective Photocatalytic Reduction of Nitrite to Nitrogen on Silver-Doped Titanium Dioxide. Journal of Physical Chemistry C, 111, 3756-3761.
http://dx.doi.org/10.1021/jp067807j

58. 58. Zhang, H. and Chen, G. (2009) Potent Antibacterial Activities of Ag/TiO2 Nanocomposite Powders Synthesized by a One-Pot Sol-Gel Method. Environmental Science and Technology, 43, 2905-2910.
http://dx.doi.org/10.1021/es803450f

59. 59. Zhang, T., Oyama, T., Aoshima, A., Hidaka, H., Zhao, J. and Serpone, N. (2001) Photooxidative N-Demethylation of Methylene Blue in Aqueous TiO2 Dispersions under UV Irradiation. Journal of Photochemistry and Photobiology A: Chemistry, 140, 163-172.
http://dx.doi.org/10.1016/S1010-6030(01)00398-7

60. 60. Zhao, X.F., Meng, X.F., Zhang, Z.H., Liu, L. and Jia, D.Z. (2004) Preparation and Photocatalytic Activity of Pb-Doped TiO2 Thin Films. Journal of Inorganic Materials, 19, 140-146.

61. 61. Dobosz, A. and Sobczyński, A. (2003) The Influence of Silver Additives on Titaniaphotoactivity in the Photooxidation of Phenol. Water Resources, 37, 1489-1496.

62. 62. Dvoranová, D., Brezová, V., Mazúr, M. and Malati, M.A. (2002) Investigations of Metal-Doped Titanium Dioxide Photocatalysts. Applied Catalysis B: Environmental, 37, 91-105.
http://dx.doi.org/10.1016/S0926-3373(01)00335-6

63. 63. Behpour, M., Ghoreishi, S.M. and Razavi, F.S. (2010) Photocatalytic Activity of TiO2/Ag Nanoparticle on Degradation of Water Pollutions. Digest Journal of Nanomaterial Biostructures, 5, 467-475.

64. 64. Fatunla, S.O. (1987) An Implicit Two-Point Numerical Integration Formula for Linear and Non-Linear Stiff System of ODEs. Mathematics of Computation, 32, 1-11.
http://dx.doi.org/10.1090/S0025-5718-1978-0474830-0

65. 65. Ibijola, E.A. (1997) A New Numerical Scheme for the Solution of Initial Value Problem (IVPs). Ph.D. Thesis, University of Benin, Nigeria.

66. 66. Ibijola, E.A. (1998) On the Convergence, Consistency and Stability of a One-Step Method for Integration of ODEs. International Journal of Computer Mathematics, 73, 261-277.

67. 67. Ogunrinde, R.B. (2010) A New Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations. Ph.D. Thesis, University of Ado Ekiti, Nigeria.

68. 68. Obayomi, A.A. (2012) Derivation of Non-Standard Finite Difference Schemes for the Second Order Chemical Reaction Model. Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine, 3, 121-124.

69. 69. Obayomi, A.A. (2012) A Set of Non-Standard Finite Difference Schemes for the Solution of an Equation of the Type International Journal of Pure and Applied Sciences and Technology, 12, 34-42.