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On the High-Order Quasi Exactly Solvable Differential Equations
Department of Electronic Engineering, College of Technological Studies, PAAET, Kuwait City, Kuwait
Department of Applied Sciences, College of Technological Studies, PAAET, Kuwait City, Kuwait
Department of Applied Sciences, College of Technological Studies, PAAET, Kuwait City, Kuwait
- 1 Department of Electronic Engineering, College of Technological Studies, PAAET, Kuwait City, Kuwait
- 2 Department of Applied Sciences, College of Technological Studies, PAAET, Kuwait City, Kuwait
- 3 Department of Applied Sciences, College of Technological Studies, PAAET, Kuwait City, Kuwait
American Journal of Computational Mathematics·Volume 09 (2019)·Pages 234–250·Published 25 October 2019·DOI10.4236/ajcm.2019.94018
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Abstract
In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.
KeywordsQuasi-Exactly Solvable High-Order Differential EquationsCanonical PolynomialsTau Method
- Schafke, R. and Schmidt, D. (1980) The Connection Problem for General Linear Ordinary Differential Equations at Two Regular Singular Points with Applications in the Theory of Special Functions. SIAM Journal on Mathematical Analysis, 11, 848-862. https://doi.org/10.1137/0511076
- Bender, C.M. and Dunne, G.V. (1996) Quasi-Exactly Solvable Systems and Orthogonal Polynomials. Journal of Mathematical Physics, 37, 6-11. https://doi.org/10.1063/1.531373
- Sasaki, R., Yang, W.L. and Zhang, Y.Z. (2009) Bethe Ansatz Solutions to Quasi-Exactly Solvable Difference Equations. Symmetry, Integrability and Geometry: Methods and Applications, 5, 104. https://doi.org/10.3842/SIGMA.2009.104
- Gomez-Ullate, D., Kamran, N. and Milson, R. (2009) An Extended Class of Orthogonal Polynomials Defined by a Sturm-Liouville Problem. Journal of Mathematical Analysis and Applications, 359, 352-367. https://doi.org/10.1016/j.jmaa.2009.05.052
- Pan, F., Klauder, J.R. and Draayer, J.P. (1999) Quasi-Exactly Solvable Cases of an N-Dimensional Symmetric Decatic Anharmonic Oscillator. Physics Letters A, 262, 131-136. https://doi.org/10.1016/S0375-9601(99)00651-9
- Moroz, A. and Miroshnichenko, A.E. (2018) Constraint Polynomial Approach—An Alternative to the Functional Bethe Ansatz Method? arXiv:1807.11871v1.
- Xie, Q.-T. (2012) New Quasi-Exactly Solvable Double-Well Potentials. Journal of Physics A: Mathematical and Theoretical, 45, Article ID: 175302. https://doi.org/10.1088/1751-8113/45/17/175302
- El-Daou, M.K., AlZanki, T.H. and Al-Mutawa, N.S. (2019) Quasi-Exactly Solvable Differential Models: A Canonical Polynomials Approach. American Journal of Computational Mathematics, 9, 48-60. https://doi.org/10.4236/ajcm.2019.92004
- El-Daou, M.K. (2012) Duality between the Generalized Canonical Polynomials and Some Special Determinants. International Journal of Applied Mathematics, 25, 811-824.
- Lanczos, C. (1956) Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ.
- Ortiz, E.L. (1969) The Tau Method. SIAM Journal on Numerical Analysis, 6, 480-492. https://doi.org/10.1137/0706044
- Froes Bunchaft, M.E. (1997) Some Extensions of the Lanczos-Ortiz Theory of Canonical Polynomials in the Tau Method. Mathematics of Computation, 66, 609-621. https://doi.org/10.1090/S0025-5718-97-00816-8
- El-Daou, M.K. and Ortiz, E.L. (1994) A Recursive Formulation of Collocation in Terms of Canonical Polynomials. Computing, 52, 177-202. https://doi.org/10.1007/BF02238075