A Perturbative-Based Generalized Series Expansion in Terms of Non-Orthogonal Component Functions

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DOI: 10.4236/am.2017.81009    398 Downloads   550 Views  

ABSTRACT

In this paper we present a generalized perturbative approximate series expansion in terms of non-orthogonal component functions. The expansion is based on a perturbative formulation where, in the non-orthogonal case, the contribution of a given component function, at each point, in the time domain or frequency in the Fourier domain, is assumed to be perturbed by contributions from the other component functions in the set. In the case of orthogonal basis functions, the formulation reduces to the non-perturbative case approximate series expansion. Application of the series expansion is demonstrated in the context of two non-orthogonal component function sets. The technique is applied to a series of non-orthogonalized Bessel functions of the first kind that are used to construct a compound function for which the coefficients are determined utilizing the proposed approach. In a second application, the technique is applied to an example associated with the inverse problem in electrophysiology and is demonstrated through decomposition of a compound evoked potential from a peripheral nerve trunk in terms of contributing evoked potentials from individual nerve fibers of varying diameter. An additional application of the perturbative approximation is illustrated in the context of a trigonometric Fourier series representation of a continuous time signal where the technique is used to compute an approximation of the Fourier series coefficients. From these examples, it will be demonstrated that in the case of non-orthogonal component functions, the technique performs significantly better than the generalized Fourier series which can yield nonsensical results.

Cite this paper

Szlavik, R. , Paquin, D. and Turner III, G. (2017) A Perturbative-Based Generalized Series Expansion in Terms of Non-Orthogonal Component Functions. Applied Mathematics, 8, 106-116. doi: 10.4236/am.2017.81009.

References

[1] Lomen, D. and Mark, J. (1988) Differential Equations. Prentice-Hall, Englewood Cliffs.
[2] Kreyszig, E. (1988) Advanced Engineering Mathematics, Vol. 6. John Wiley & Sons, New York.
[3] Harati, Y. (1987) Diabetic Peripheral Neuropathies. Annals of Internal Medicine, 107, 546-559.
https://doi.org/10.7326/0003-4819-107-4-546
[4] Dorfman, L., Cummins, K., Reaven, G., Ceranski, J., Greenfield, M. and Doberne, L. (1983) Studies of Diabetic Polyneuropathy Using Conduction-Velocity Distribution (dcv) Analysis. Neurology, 33, 773-779.
https://doi.org/10.1212/WNL.33.6.773
[5] Szlavik, R. (2016) A Perturbation Based Decomposition of Compound Evoked Potentials for Characterization of Nerve Fiber Size Distributions. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 24, 212-216.
https://doi.org/10.1109/TNSRE.2015.2476917
[6] Szlavik, R. and de Bruin, H. (1997) Simulating the Distribution of Axon Size in Nerves. CMBES/SCGB, 168-169.
[7] Boyd, I.A. and Davey, M.R. (1968) Composition of Peripheral Nerves. E. & S. Livingstone, Edinburgh.
[8] Fleisher, S., Studer, M. and Moschytz, G. (1984) Mathematical-Model of the Single-Fiber Action-Potential. Medical & Biological Engineering & Computing, 22, 433-439.
https://doi.org/10.1007/BF02447703
[9] Szlavik, R. (2008) A Novel Method for Characterization of Peripheral Nerve Fiber Size Distributions by Group Delay. IEEE Transactions on Biomedical Engineering, 55, 2836-2840.
https://doi.org/10.1109/TBME.2008.921149
[10] Sinha, N.K. (1991) Linear Systems. John Wiley & Sons, New York.

  
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