On the Solutions of Schlömilch's Integral Equations

Abstract

The linear Schlömilch's integral equation is an important and useful equation in atmospheric and terrestrial physics. The equation and its solution have been used for some ionospheric problems. It can also be considered as a special type of Fredholm integral equation of the first kind. This correspondence allows one to use the mathematical tools available for solving Fredholm integral equation of the first kind. In this article, we provide an alternative closed-form expression for solutions of the linear and the nonlinear Schlömilch's integral equation in terms of the well-known gamma function. Some elaborate examples are provided to demonstrate the simplicity and applicability of the proposed formulae.

Keywords

• Schlömilch’s Integral Equation,Fredholm Integral Equation of the First Kind,Terrestrial Physics,Gamma Function

References

1. 1. Unz, H, Schlömilch’s Integral Equation, Journal of Atmosphe-ric and Terrestrial Physics, 1963, 25, 101–102.
2. 2. Unz, H, Schlömilch’s Integral Equation for Oblique Incidence, Journal of Atmospheric and Terrestrial Physics, 1966, 28, 315–316.
3. 3. Gething, P.J.D, Maliphant, R.G, Unz’s Application of Schlo-milch’s Integral Equation to Oblique Incidence Observations, Journal of Atmospheric and Terrestrial Physics, 1967, 29, 599–600.
4. 4. Bougoffa, L, Al-Hagbani, M, Brceski, I, Randolph, C.R.A, Convenient Technique for Solving Integral Equations of the First Kind by the Adomian Decomposition Method, Kybernetes, 2012, 41, 145-156.
5. 5. Parand, K, Delkosh, M, Solving the Nonlinear Schlomilch’s Integral Equation Arising in Ionospheric Problems, Afrika Matemat-ika, 2016, doi: 10.1007/s13370-016-0459-3.
6. 6. Wazwaz, A, Solving Schlömilch’s Integral Equation by the Regularization-Adomian Method, Romanian Journal of Physics, 2015, 60, 56 – 71.
7. 7. De, S.S, Sarkar, B.K, Manasi, M, De, M, Gosh, B, Adhikari, S.K, On Schlomilch’s Integral Equation for the Ionospheric Plas-ma, Japanese Journal of Applied Physics, 1994, 33, 1-7A.
8. 8. Tikhonov, A.N, Solution of Incorrectly Formulated Problems and the Regularization Method, Soviet Mathematics Doklady, 1963, 4, 1035-1038.

9. 9. Tikhonov, A.N, Regularization of Incorrectly Posed Problems, Soviet Mathematics Doklady, 1963, 4, 1624-1627.
10. 10. Philips, D.L. A, Technique for the Numerical Solution of Cer-tain Integral Equations of the First Kind, Journal of the Associa-tion for Computing Machinery, 1962, 84-96.
11. 11. Adomian, G, Solving Frontier Problems of Physics, the De-composition Method, 1994; Kluwer, Boston.
12. 12. Wazwaz, A, Linear and Nonlinear Integral Equations: Methods and Applications; Springer and Hep: Berlin and Beijing, 2011; pp 658.
13. 13. Wade, W.R, An Introduction to Analysis; Pearson Prentice Hall: New Jersey, 2010; pp 680.
14. 14. Wastlund, J, An Elementary Proof of the Wallis Product Formu-la for Pi. The mathematical association of America, 2007; 114, 914-917.
15. 15. Khrushchev, S, A Recovery of Brouncker’s Proof for the Quad-rature Continued Fraction, Publicacions Matematiques, 2006, 50, 3-42.