We consider a broad class of linear operator equations that includes systems of ordinary differential equations, difference equations and fractional-order ordinary differential equations. This class also includes operator exponentials and powers, as well as eigenvalue problems and Fredholm integral equations. Many problems in engineering and the physical and natural sciences can be described by such operator equations. We generalise the fundamental matrix to a fundamental operator and provide a new explicit method for obtaining an exact series solution to these types of operator equations, together with sufficient conditions for convergence and error bounds. Illustrative examples are also given.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 3
Year: 2020

DOI: 10.12732/ijam.v33i3.5

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  1. [1] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York (2012).
  2. [2] F. Brauer, J.A. Nohel, The Qualitative Theory of Ordinary Differential Equations, Dover, New York (1989).
  3. [3] G. de Vries, T. Hillen, M. Lewis, J. M¨uller, B. Sch¨ofisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods, SIAM, Philadelphia (2006).
  4. [4] S.N. Elaydi, An Introduction to Difference Equations, Springer, New York (2000).
  5. [5] S.N. Elaydi, W.H. Harris, Jr., On the computation of AN, SIAM Rev., 40, No 4 (1998), 965-971.
  6. [6] D.H. Griffel, Applied Functional Analysis, Wiley & Sons, New York (1981).
  7. [7] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).
  8. [8] C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, No 1 (2003), 3-49.
  9. [9] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering 198, Academic Press, San Diego (1999).
  10. [10] E.J. Putzer, Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients, Amer. Math. Monthly, 73, No 1 (1966), 2-7.
  11. [11] M.R. Rodrigo, On fractional matrix exponentials and their explicit calculation, J. Differential Equations, 261, No 7 (2016), 4223-4243.
  12. [12] H. Sch¨afer, E. Sternin, R. Stannarius, M. Arndt, F. Kremer (1996), Novel approach to the analysis of broadband dielectric spectra, Phys. Rev. Lett., 76, No 12 (1996), 2177-2180.
  13. [13] D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., 25 No 3 (1983), 379-393.
  14. [14] A.M. Teguia, Extensions of the Cayley-Hamilton Theorem with applications to elliptic operators and frames, Electronic Theses and Dissertations, Paper 1024 (2005), http://dc.etsu.edu/etd/1024.
  15. [15] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Applied Mathematical Sciences Series, Vol. 108, Springer-Verlag, New York (1995).