# Transformed Hermite functions on a finite interval and their applications to a class of singular boundary value problems

Article

First Online:

Received:

Revised:

Accepted:

- 100 Downloads
- 1 Citations

## Abstract

In this paper, a weighted orthogonal system on finite interval based on the transformed Hermite functions is introduced. Some results on approximations using the Hermite functions on finite interval are obtained from corresponding approximations on infinite interval via a conformal map. To illustrate the potential of the new basis, we apply it to the collocation method for solving a class of singular two-point boundary value problems. The numerical results show that our new scheme is very effective and convenient for solving singular boundary value problems.

### Keywords

Transformed Hermite functions Collocation Conformal map Singular boundary value problems### Mathematics Subject Classification

Primary 65L60 Secondary 34B16## Notes

### Acknowledgments

The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

### References

- Adams RA (1975) Sobolev spaces. Academic Press, New YorkMATHGoogle Scholar
- Al-Khaled K (2001) Singular two-point boundary value problems. Lib Math 21:75–82MathSciNetMATHGoogle Scholar
- Al-Khaled K (2007) Theory and computation in singular boundary value problems. Chaos Solit Fract 33:678–684MathSciNetCrossRefMATHGoogle Scholar
- Babolian E, Eftekhari A, Saadatmandi A (2014) A sinc-Galerkin approximate solution of the reaction–diffusion process in an immobilized biocatalyst pellet. MATCH Commun Math Comput Chem 71:681–697MathSciNetGoogle Scholar
- Bao WZ, Shen J (2005) A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates. SIAM J Sci Comput 26:2010–2028MathSciNetCrossRefMATHGoogle Scholar
- Bataineh AS, Noorani MSM, Hashim I (2008) Approximate solutions of singular two-point BVPs by modified homotopy analysis method. Phys Lett A 372:4062–4066CrossRefMATHGoogle Scholar
- Beliczynski B (2011) Approximation of functions by multivariable Hermite basis: a hybrid method. Lect Notes Comput Sci 6593:130–139CrossRefGoogle Scholar
- Beliczynski B (2012) A method of multivariable Hermite basis function approximation. Neurocomputing 96:12–18CrossRefGoogle Scholar
- Boyd JP (1980) The rate of convergence of Hermite function series. Math Comput 35:1309–1316MathSciNetCrossRefMATHGoogle Scholar
- Boyd JP (1984) Asymptotic coefficients of Hermite function series. J Comput Phys 54:382–410MathSciNetCrossRefMATHGoogle Scholar
- Boyd JP (1987) Orthogonal rational functions on a semi-infinite interval. J Comput Phys 70:63–88MathSciNetCrossRefMATHGoogle Scholar
- Boyd JP (2001) Chebyshev and Fourier spectral methods, 2nd edn. Dover, NewYorkMATHGoogle Scholar
- Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods: fundamentals in single domains. Springer, New YorkMATHGoogle Scholar
- Doha EH, Bhrawy AH, Abdelkawy MA, Hafez RM (2014) A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation. Cent Eur J Phys 12:111–122Google Scholar
- Doha EH, Bhrawy AH, Baleanu D, Hafez RM (2014) A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations. Appl Numer Math 77:43–54MathSciNetCrossRefMATHGoogle Scholar
- Dym H, McKean HP (1976) Gaussian processes, function theory and the inverse spectral problem. Academic Press, New YorkMATHGoogle Scholar
- Guo BY, Shen J, Wang ZQ (2000) A rational approximation and its applications to differential equations on the half line. J Sci Comput 15:117–147MathSciNetCrossRefMATHGoogle Scholar
- Guo BY, Shen J, Xu CL (2003) Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation. Adv Comput Math 19:35–55MathSciNetCrossRefMATHGoogle Scholar
- Kadalbajoo MK, Aggarwal VK (2004) Cubic spline for solving singular two-point boundary value problems. Appl Math Comput 156:249–259MathSciNetMATHGoogle Scholar
- Kanth ASVR, Aruna K (2008) Solution of singular two-point boundary value problems using differential transformation method. Phys Lett A 372:4671–4673MathSciNetCrossRefMATHGoogle Scholar
- Kumar M (2003) A second order spline finite difference method for singular two-point boundary value problems. Appl Math Comput 142:283–290MathSciNetMATHGoogle Scholar
- Lu J (2007) Variational iteration method for solving two-point boundary value problems. J Comput Appl Math 207:92–95MathSciNetCrossRefMATHGoogle Scholar
- Luo X, Yau SST (2013) Hermite spectral method to 1-D forward Kolmogorov equation and its application to nonlinear filtering problems. IEEE Trans Autom Control 58:2495–2570MathSciNetCrossRefGoogle Scholar
- Parand K, Dehghan M, Rezaei AR, Ghaderi SM (2010) An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method. Comput Phys Commun 181:1096–1108MathSciNetCrossRefMATHGoogle Scholar
- Saadatmandi A, Razzaghi M, Dehghan M (2005) Sinc-collocation methods for the solution of Hallen’s integral equation. J Electromagn Waves Appl 19:245–256MathSciNetCrossRefGoogle Scholar
- Saadatmandi A, Nafar N, Toufighi SP (2014) Numerical study on the reaction cum diffusion process in a spherical biocatalyst. Iran J Math Chem 5:47–61Google Scholar
- Saadatmandi A, Dehghan M (2008) A collocation method for solving Abel’s integral equations of first and second kinds. Zeitschrift für Naturforschung A 63:752–756CrossRefGoogle Scholar
- Sababheh MS, Nusayr AM, Al-Khaled K (2003) Some convergence results on sinc interpolation. J Inequal Pure Appl Math 4:32–48MathSciNetMATHGoogle Scholar
- Shen J, Tang T, Wang L-L (2010) Spectral methods, algorithms, analysis and applications. Springer, New YorkGoogle Scholar
- Shen J, Wang LL (2009) Some recent advances on spectral methods for unbounded domains. Commun Comput Phys 5:195–241MathSciNetGoogle Scholar
- Shin JY (1995) A singular nonlinear boundary value problem in the nonlinear circular membrance under normal pressure. J Kor Math Soc 32:761–773MathSciNetMATHGoogle Scholar
- Tang T (1993) The Hermite spectral method for Gauss-type function. SIAM J Sci Comput 14:594–606MathSciNetCrossRefMATHGoogle Scholar
- Xiang XM, Wang ZQ (2010) Generalized Hermite spectral method and its applications to problems in unbounded domains. SIAM J Numer Anal 48:1231–1253MathSciNetCrossRefMATHGoogle Scholar
- Xiang XM, Wang ZQ (2013) Generalized Hermite approximations and spectral method for partial differential equations in multiple dimensions. J Sci Comput 57:229–253MathSciNetCrossRefMATHGoogle Scholar
- Yeganeh S, Saadatmandi A, Soltanian F, Dehghan M (2013) The numerical solution of differential-algebraic equations by sinc-collocation method. Comput Appl Math 32:343–354MathSciNetCrossRefMATHGoogle Scholar
- Yin Z (2014) A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains. J Comput Phys 258:371–380MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015