The Multiplicity of Eigenvalues of a Vectorial Diffusion Equations with Discontinuous Function Inside A Finite Interval

Abstract

In this study, diffusion operator with discontinuity function is considered. Integral equations have been obtained for the solution under certain initial conditions. Furthermore, we obtained integral representations for these solutions. Some equations have been obtained by the kernel functions. By using the characteristic function, asymptotic formulas of eigenvalues with zeros of the characteristic are obtained.       

In this study, diffusion operator with discontinuity function is considered. Integral equations have been obtained for the solution under certain initial conditions. Furthermore, we obtained integral representations for these solutions. Some equations have been obtained by the kernel functions. By using the characteristic function, asymptotic formulas of eigenvalues with zeros of the characteristic are obtained.     

Keywords

• Diffrential equations
• Vectorial Diffusion operator

References

1. [1] Agranovich ZS, MarchencoVA. The inverse problem of scattering theory. Gordon and Breach Science Publisher.NewYork-London. 1963.
2. [2] Bellmann R. Introduction to matrix analysis (2nd ed.) McGraw-Hill. 1970.
3. [3] Boas RP. Entire functions. Academic press. New York. 1954.
4. [4] Kong Q. Multiplicities of eigenvalues of a vector-valued Sturm-Liouville Problem. Mathematica. 49(1-2), 2002, 119-127.
5. [5] Shen CL, Shieh C. On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equations and some related spectral problems. Proc. Amer. Math. Soc. 127(10), 1999, 2943-2952.
6. [6] Yang CF, Huang ZY, Yang XP. The multiplicity of spectra of a vectorial Sturm- Liouville differential equation of dimension two and some applications. Rock mountain journal of Mathematics.37(4) 2007, 1379-1398.
7. [7] Amirov RK. On Sturm-Liouville operators with discontiniuity conditions inside an interval. Journal of Mathematical Analysis and Aplications. 317(1), 2006, 163-176.
8. [8] Carvert JM, Davison WD. Oscillation theory and computational procedures for matrix Sturm-Liouville eigenvalue problems with an application to the hydrogen molecular ion. Journal of Physics A Mathematical and General. 2(3), 1969, 278-292.

9. [9] Müller G. The reflectivity method a tutorial. J. Geophys. 58, 1985, 153-174.
10. [10] Kauuffman RM, Zhang HK. A class of ordinary differential operators with jump baundary conditions. Lecture notes in Pure and Appl. Math.234, 2003, 253-274.
11. [11] Mukhtarov O, Yakubov S. Problem for ordinary differential equations with transmission conditions. Appl. Anal, 81(5), 2002, 1033-1064.
12. [12] Wang AP, Sun J, Zettl,A. Two-interval Sturm-Liouville operators in modified Hilbert spaces. Journal of Mathematical Analysis and Applications. 328(1), 2007, 390-399.
13. [14] Gasymov MG, Guseinov GSh. Determination diffusion operator on spectral data, SSSR Dokl. 37(2), 1981, 19-23.
14. [15] Koyunbakan H, Panakhov ES. Half inverse problem for diffusion operators on the finite interval. J. Math. Anal. Appl. 326, 2007, 1024-1030.
15. [16] Ergun A, Amirov RKh. Direct and Inverse problem for Diffusion operator with Discontinuity points. TWMS J. App. Eng. Math. 9, 2019, 9-21.
16. [17] Yurko VA. Introduction to the Theory of Inverse Spectral Problems. Russian. Fizmatlit. 2007.
17. [18] Levitan BM, Sargsyan IS. Introduction to Spectral Theory. Amer. Math. Soc. 1975.
18. [19] Marchenko VA. Sturm-Liouville Operators and Applications. AMS. Chelsea Publishing. 1986.
19. [20] Titchmarsh EC. The Theory of Functions. Oxford at the clarendon press London. 1932.
20. [21] Levitan BM. Inverse Sturm-Liouville Problems. Engl.Transl. NU Science Press. 1987.
21. [22] Vakanas LP.Ascattering parameter based method for the transient analysis of lossy, coupled, nonlinearly terminated transmission line systems in high-speed microelectronic circuits. IEEE Transactions on Components Packaging and Manufacturing Technology Part B. 17(4), 1994, 472-479.
22. [23] Beck JV. Inverse Heat Conduction. John Wiley and Sons New York. 1985.
23. [24] Kirkup SM, Wadsworth M. Computational solution of the atomic mixing equations: special methods and algorithm of impetus II. International Journal of Numerical Modelling. Electronic Networks Devices and Fields. 11, 1998, 207-219.