## American Journal of Mathematical Analysis

**Current Issue» **Volume 2, Number 3 (2014)

## Article

# He-Laplace Method for the Solution of Two-point Boundary Value Problems

^{1}Department of Mathematics, Jaypee University of Engineering & Technology, Guna-473226(M.P), India

*American Journal of Mathematical Analysis*.

**2014**, 2(3), 45-49

**DOI:**10.12691/ajma-2-3-3

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Hradyesh Kumar Mishra. He-Laplace Method for the Solution of Two-point Boundary Value Problems.

*American Journal of Mathematical Analysis*. 2014; 2(3):45-49. doi: 10.12691/ajma-2-3-3.

Correspondence to: Hradyesh Kumar Mishra, Department of Mathematics, Jaypee University of Engineering & Technology, Guna-473226(M.P), India. Email: hk.mishra@juet.ac.in

## Abstract

## Keywords

## References

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[1] | G.Adomian, A revieal of the decomposition method and some recent results for nonlinear equation, Mathematical and Computer Modeling 13(7)(1992) 17-43. | ||

[2] | G.Adomian, A revieal of the decomposition method in applied mathematics, Journal of Mathematical Analysis and application 135(1988) 501-544. | ||

[3] | G.Adomian, Solution frontier Problems of Physics: The Decomposition method, Kluwer Dordrecht, 1994. | ||

[4] | C.Chun, R.Sakthivel, Homotopy perturbation technique for solving two point boundary value problems- comparison with other methods, Computer physics communications, 181(2010) 1021-1024. | ||

[5] | M.Dehghan, A Saadatmandi, the numerical solution of a nonlinear system of second order boundary value problems using the Sinc-Collocation method, Mathematical and Computer Modeling 46(2007) 1434-1441. | ||

[6] | S.N. Ha, A nonlinear shooting method for two-point boundary value problems, Computers and Mathematics with Applications 42(2001) 1411-1420. | ||

[7] | J.H.He, Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation, Vol. 135, 2003, pp.73-79. | ||

[8] | J.H.He, Homotopy, Comparion of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, Vol. 156, 2004, pp. 527-539. | ||

[9] | J.H.He, Homotopy, The homotopy perturbation method for nonlinear oscillators with discontinueties, Applied Mathematics and Computation, Vol. 151, 2004, pp. 287-292. | ||

[10] | J.H.He, Recent developments of the homotopy perturbation method, Topological methods in Non-linear Analysis, Vol. 31, 2008, pp. 205-209. | ||

[11] | J.H.He, New Interpretation of homotopy perturbation method, International Journal of Modern Physics, Vol. 20, 2006, pp. 2561-2568. | ||

[12] | J.H.He, A coupling method of homotopy technique and a perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics, Vol. 35, 2000, pp. 37-43. | ||

[13] | J.H.He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, Vol. 114, 2000, pp. 115-123. | ||

[14] | J.H.He, Homotopy perturbation technique, Computer methods in Applied Mechanics and Engineering, Vol. 178, 1999, pp. 257-262. | ||

[15] | J.H.He, A simple perturbation approach to Blasius equation, Applied Mathematics and Computation, Vol.140, 2003, pp. 217-222. | ||

[16] | J.H.He, Application of homotopy perturbation method to nonlinear wave equation, Chaos, Solitons, Fractals, Vol. 26, 2005, pp. 295-300. | ||

[17] | J.H.He, Homotopy perturbation method for solving boundary value problem, Physics Letter A, Vol. 350, 2006, pp. 87-88. | ||

[18] | M.T.Heath, Scientific Computing; An Introductory Survey, McGraw-Hill, New York 2002. | ||

[19] | E. Hesameddini, H. Latifizadeh, An optimal choice of initial solutions in the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, 2009, pp.1389-1398. | ||

[20] | E. Hesameddini, H. Latifizadeh, A new vision of the He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, Vol.10, 2009, pp.1415-1424. | ||

[21] | H.K. Mishra, A.K. Nagar, He-Laplace method for linear and nonlinear partial differential equations, Journal of Applied Mathematics Vol. 2012,2012, pp.1-16. | ||

[22] | H.K. Mishra, He-Laplace Method for Special Nonlinear Partial Differential Equations, Mathematical Theory and Modeling Vol.3 (6), 2013, pp. 113-117. | ||

[23] | S. Islam, Y. Khan, N. Faraz, F. Austin, Numerical solution of logistic differential equations by using the Laplace decomposition method, World Applied Sciences Journal, Vol. 8, 2010, pp. 1100-1105. | ||

[24] | B.Jang, Two point boundary value problems by the extended Adomian decomposition method, Journal of Computational and Applied Mathematics 219(2008)253-262. | ||

[25] | K.Keller, Numerical solution of two-point boundary value problems, SIAM, Philadelphia, 1976. | ||

[26] | Y.Khan, F.Austin, Application of the Laplace decomposition method to nonlinear homogeneous and non-homogeneous advection equations, Zeitschrift fuer Naturforschung, Vol. 65 a, 2010, pp.1-5. | ||

[27] | Yasir Khan, Qingbiao Wu, Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers and Mathematics with Applications, Vol.61, 2011, pp.1963-1967. | ||

[28] | Yasir khan, An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, 2009, pp. 1373-1376. | ||

[29] | S.A.Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics,Vol.1, 2001, pp.141-155. | ||

[30] | M.Madani, M.Fathizadeh, Homotopy perturbation algorithm using Laplace transformation, Nonlinear Science Letters A, Vol. 1, 2010, pp.263-267. | ||

[31] | A.Mohsen, M.El-Gamel, On the Galerkin and Collocation methods for two-point boundary value problems using Sinc bases, Computers and Mathematics with applications 56(2008)930-941. | ||

[32] | M.Rafei,D.D.Ganji, Explicit solutions of helmhotz equation and fifth-order KdV equation using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, 2006, pp. 321-328. | ||

[33] | A.M.Siddiqui, R.Mohmood,Q.K.Ghori,Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, 2006, pp. 7-14. | ||

[34] | O.A. Taiwo, Exponential fitting for the solution of two-point boundary value problems with cubic spline collocation tau method International Journal of Computer Mathematics, 79(2002)299-306. | ||

[35] | A.M. Waz Waz, A reliable modification of Adomian’s decomposition method, Applied Mathematics and computation 92(1998)1-3. | ||

[36] | A.M.Waz Waz, A composition between Adomian’s decomposition method and Taylor series method in the series solution, Applied Mathematics and computation 79(1998)37-44. | ||

[37] | A.M.Waz Waz, Adomian decomposition method for a reliable treatment of the Bratu type equations, Applied Mathematics and computation 166(2005)652-663. | ||

[38] | A.M.Waz Waz, Partial Differential Equations; Methods and Applications, Balkema Publishers, The Netherland, 2002. | ||

[39] | L.Xu, He’s homotopy perturbation method for a boundary layer equation in unbounded domain, Computers and Mathematics with Applications, vol. 54, 2007, pp. 1067-1070. | ||

## Article

# Modular Relations for the Sextodecic Analogues of the Rogers-Ramanujan Functions with its Applications to Partitions

^{1}Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore, India

*American Journal of Mathematical Analysis*.

**2014**, 2(3), 36-44

**DOI:**10.12691/ajma-2-3-2

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Adiga Chandrashekar, Nasser Abdo Saeed Bulkhali. Modular Relations for the Sextodecic Analogues of the Rogers-Ramanujan Functions with its Applications to Partitions.

*American Journal of Mathematical Analysis*. 2014; 2(3):36-44. doi: 10.12691/ajma-2-3-2.

Correspondence to: Adiga Chandrashekar, Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore, India. Email: c_adiga@hotmail.com

## Abstract

## Keywords

## References

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[1] | C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan’s second notebook: Theta functions and q -series, Mem. Amer. Math. Soc. 315 (1985), 1-91. | ||

[2] | C. Adiga and N. A. S. Bulkhali, Some modular relations analogues to the Ramanujan’s forty identities with its applications to partitions, Axioms, 2(1) (2013), 20-43. | ||

[3] | C. Adiga and N. A. S. Bulkhali, On certain new modular relations for the Rogers-Ramanujan type functions of order ten and its applications to partitions, Note di Matematica, to appear. | ||

[4] | C. Adiga and M. S. Surekha, Some modular relations for the Roger-Ramanujan type functions of order six and its applications to partitions, Proc. J. math. Soc., to appear. | ||

[5] | C. Adiga and M. S. Surekha and N. A. S. Bulkhali, Some modular relations of order six with its applications to partitions, Gulf J. Math., to appear. | ||

[6] | C. Adiga and A. Vanitha, New modular relations for the Rogersâ€“Ramanujan type functions of order fifteen, Notes on Numb. Theor. Discrete Math., 20(1) (2014), 36-48. | ||

[7] | C. Adiga and A. Vanitha and N. A. S. Bulkhali, Modular relations for the Rogers-Ramanujan-Slater type functions of order fifteen and its applications to partitions, Romanian J. Math. Comput. Sci., 3(2) (2013), 119-139. | ||

[8] | C. Adiga and A. Vanitha and N. A. S. Bulkhali, Some modular relations for the Rogers-Ramanujan type functions of order fifteen and its applications to partitions, Palestine J. Math., 3(2) (2014), 204-217. | ||

[9] | C. Adiga, K. R. Vasuki and N. Bhaskar, Some new modular relations for the cubic functions, South East Asian Bull. Math., 36 (2012), 1-19. | ||

[10] | C. Adiga, K. R. Vasuki and B. R. Srivatsa Kumar, On modular relations for the functions analogous to Rogers-Ramanujan functions with applications to partitions, South East J. Math. and Math. Sc., 6(2) (2008), 131-144. | ||

[11] | G. E. Andrews and B. C. Berndt, Ramanujans Lost Notebook, Part III, Springer, New York, 2012. | ||

[12] | N. D. Baruah and J. Bora, Further analogues of the Rogers-Ramanujan functions with applications to partitions, Elec. J. Combin. Number Thy., 7(2) (2007), ＃ A05, 22pp. | ||

[13] | N. D. Baruah and J. Bora, Modular relations for the nonic analogues of the Rogers-Ramanujan functions with applications to partitions, J. Number Thy., 128 (2008), 175-206. | ||

[14] | N. D. Baruah, J. Bora and N. Saikia, Some new proofs of modular relations for the Göllnitz-Gordon functions, Ramanujan J., 15 (2008), 281-301. | ||

[15] | B. C. Berndt and H. Yesilyurt, New identities for the Rogers-Ramanujan function, Acta Arith., 120 (2005), 395-413. | ||

[16] | B. C. Berndt, G. Choi, Y. S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt, and J. Yi, Ramanujan’s forty identities for the Rogers-Ramanujan function, Mem., Amer. Math. Soc., 188(880) (2007), 1-96. | ||

[17] | A. J. F. Biagioli, A proof of some identities of Ramanujan using modular functions, Glasg. Math. J., 31 (1989), 271-295. | ||

[18] | B. J. Birch, A look back at Ramanujan’s Notebooks, Math. Proc. Camb. Soc., 78 (1975), 73-79. | ||

[19] | R. Blecksmith. J. Brillhat. and I. Gerst. A foundamental modular identity and some applications. Math. Comp., 61 (1993), 83-95. | ||

[20] | D. Bressoud, Proof and Generalization of Certain Identities Conjectured by Ramanujan, Ph.D. Thesis, Temple University, 1977. | ||

[21] | D. Bressoud, Some identities involving Rogers-Ramanujan-type functions, J. London Math. Soc., 16(2) (1977), 9-18. | ||

[22] | S. L. Chen and S.-S. Huang, New modular relations for the Göllnitz-Gordon functions, J. Number Thy., 93 (2002), 58-75. | ||

[23] | C. Gugg, Modular Identities for the Rogers-Ramanujan Function and Analogues, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2010. | ||

[24] | H. Hahn, Septic analogues of the Rogers-Ramanujan functions, Acta Arith., 110 (4) (2003), 381-399. | ||

[25] | H. Hahn, Eisenstein Series, Analogues of the Rogers-Ramanujan Functions, and Parttition Identities, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2004. | ||

[26] | S.-S. Huang, On modular relations for the Göllnitz-Gordon functions with applications to partitions, J. Number Thy., 68 (1998), 178-216. | ||

[27] | S. Ramanujan, Algebraic relations between certain infinite products, Proc. London Math. Soc., 2 (1920), p. xviii. | ||

[28] | S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. | ||

[29] | S. Robins, Arithmetic Properties of Modular Forms, Ph.D. Thesis, Univer- sity of California at Los Angeles, 1991. | ||

[30] | L. J. Rogers, On a type of modular relation, Proc. London Math. Soc., 19 (1921), 387-397. | ||

[31] | L. J. Slater, Further identities of Rogers-Ramanujan type, London Math. Soc., 54(2) (1952), 147-167. | ||

[32] | K. R. Vasuki and P. S. Guruprasad, On certain new modular relations for the Rogers-Ramanujan type functions of order twelve, Adv. Stud. Contem. Math., 20(3) (2000), 319-333. | ||

[33] | K. R. Vasuki, G. Sharath, and K. R. Rajanna, Two modular equations for squares of the cubic-functions with applications, Note di Matematica, 30(2) (2010), 61-71. | ||

[34] | G. N. Watson, Proof of certain identities in combinatory analysis, J. Indian Math. Soc., 20 (1933), 57-69. | ||

[35] | E. X. W. Xia and X. M. Yao, Some modular relations for the Göllnitz-Gordon functions by an even-odd method, J. Math. Anal. Appl., 387 (2012), 126-138. | ||

[36] | H. Yesilyurt, A generalization of a modular identity of Rogers, J. Number Thy., 129 (2009), 1256-1271. | ||

## Article

# Some Identities Involving Common Factors of k-Fibonacci and k-Lucas Numbers

^{1}School of Studies in Mathematics, Vikram University, Ujjain (India)

^{2}College of Horticulture, Mandsaur (India)

*American Journal of Mathematical Analysis*.

**2014**, 2(3), 33-35

**DOI:**10.12691/ajma-2-3-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Deepika Jhala, G.P.S. Rathore, Bijendra Singh. Some Identities Involving Common Factors of k-Fibonacci and k-Lucas Numbers.

*American Journal of Mathematical Analysis*. 2014; 2(3):33-35. doi: 10.12691/ajma-2-3-1.

Correspondence to: Deepika Jhala, School of Studies in Mathematics, Vikram University, Ujjain (India). Email: jhala.deepika28@gmail.com

## Abstract

## Keywords

## References

[[[[[[[[[1] | Benjamin, A. T. and Quinn, J. J., “Recounting Fibonacci and Lucas identies”, The College Mathematics Journal, 30 (5) (1999), 359-366. | ||

[2] | Falcon, S. & Plaza, A., “On the Fibonacci k-numbers”, Chaos, Solitons & Fractals, 32 (5), (2007), 1615-24. | ||

[3] | Falcon S. and Plaza A., The k-Fibonacci hyperbolic functions, Chaos Solitons and Fractals, 38 (2) (2008), 409-420. | ||

[4] | Falcon, S., “On the k-Lucas Numbers”, Int. J. Contemp. Math. Sciences, 6 (21), (2011), 1039-1050. | ||

[5] | Hoggatt, V.E. Jr., “Fibonacci and Lucas Numbers”, Houghton – Mifflin Co., Boston (1969). | ||

[6] | Horadam A. F., A generalized Fibonacci sequences, Mathematical Magazine. 68 (1961), 455-459. | ||

[7] | Horadam A. F. and Shanon A. G.., Generalized Fibonacci triples, American Mathmatical Monthly, 80 (1973), 187-190. | ||

[8] | Koshy, T., “Fibonacci and Lucas Numbers with Applications”, John Wiley, New York (2001). | ||

[9] | Kilic E., The Binet formula, sums and representation of generalized Fibonacci p-numbers, Eur. J. Combin, 29 (3) (2008), 701-711. | ||

[10] | Panwar, Y. K., B. Singh and Gupta, V. K., “identities of Common Factors of generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers”, Applied Mathematics and Physics, 1 (4), (2013), 126-128. | ||

[11] | Thongmoon, M., “New Identities for the Even and Odd Fibonacci and Lucas Numbers”, Int. J. Contemp. Math. Sciences, 4 (14), (2009), 671-676. | ||

[12] | Thongmoon, M., “Identities for the common factors of Fibonacci and Lucas numbers”. International Mathematical Forum, 4 (7), (2009), 303-308. | ||

[13] | Yilmaz, N., Taskara, N., Uslu,K., Yazlik, Y., On the binomial sums of k-Fibonacci and k-Lucas Sequences, American institue of Physics (AIP) Conf. Proc. | ||

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