Investigation of an Exact Solution of a Mixed Boundary Value Problem Using the Residue Method

Öz

In this study, a finite differences method is proposed for solving a mixed problem, which represents phenomena in hydrodynamics. To evaluate the approximate solution of the problem, its analytical solution is also constructed by the residue method developed by M.L. Rasulov, which is applied to find the solution of the partial differential equations containing time-dependent derivatives at boundary conditions. The formula for expansion of an arbitrary function in a series of residues of the solution of the corresponding spectral problem is used to show that the solution of the mixed boundary-value problem can be represented by the given residue formula. The use of the residual method gives an exact solution for the mixed boundary value problem, represented as a rapidly decreasing series. The derived formula makes it possible to formally establish both the existence and the uniqueness of the solution. Moreover, the derived formula provides a framework for evaluation, allowing a comparative analysis between the exact solution and the numerical approach.

Anahtar Kelimeler

• Residue method
• residue representation of the solution
• expansion formula
• boundary condition with higher order derivative

Kaynakça

1. Birkhoff, G. (1908a). On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Amer. Math. Soc. 9, 219–231.
2. Birkhoff, G. (1908b). Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9, 373–395.
3. Cauchy, A. (1827). Mémoire sur l’Application du Calcul des Résidus à la Solution des Problèmes de Physique Mathématique. Paris: Chez de Bure frères.
4. Charny, I. (1963). Underground Fluid Dynamics. Moscow-Leningrad : Gostekhizdat. [in Russian]. Fourier, J. (1822). Théorie Analytique de la Chaleur. Paris: Firmin Didot.
5. Keldysh, M. (1951). On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations. Dokl. Akad. Nauk SSSR, 77, 11–14.
6. Liouville, J. (1837). Second mémoire sur le développement des fonctions ou parties de fonctions en séries dont les divers termes sont assujettis à satisfaire à une même équation différentielle du second ordre, contenant un paramètre variable. J. Math. Pures Appl., 2, 16–35.
7. Poincaré, H. (1894). Sur les équations aux dérivées partielles de la physique mathématique. Rend. Circ. Mat. Palermo, 8, 57–155.
8. Rasulov, M.L. (1959). The residue method for solution of mixed problems for differential equations and a formula for expansion of an arbitrary vector-function in fundamental functions of a boundary problem with a parameter. Matematicheskii Sbornik, 90, 277–310.

9. Rasulov, M.L. (1963). The contour integral method and its application to the solution of multi-dimensional mixed problems for differential equations of parabolic type. Matematicheskii Sbornik, 102, 393–410.
10. Rasulov, M.L. (1967). Methods of Contour Integration. Amsterdam: North-Holland Publishing Company. Rasulov, M.A., & Sinsoysal, B. (2008). Residue method for the solution of heat equation with nonlocal boundary condition. Beykent University Journal of Science and Technology, 2, 146–158.
11. Sinsoysal, B., & Rasulov, M.A. (2008). Residue method for the solution of a 2d linear heat equation with nonlocal boundary condition. Int. J. Contemp. Math. Sciences, 3(34), 1693–1700.
12. Sinsoysal, B. (2009). Residue method for the solution of wave equation with nonlocal boundary condition. Beykent University Journal of Science and Technology, 3, 74–81.
13. Sinsoysal, B., & Rasulov, M.A. (2013). Residue solution of system of differential equations and its application to RC circuit problem. Beykent University Journal of Science and Technology, 6, 1–9.
14. Sinsoysal, B., & Rasulov, M.A. (2020). One method to prove of existence weak solution of a mixed problem for 2D parabolic equations. Partial Differential Equations in Applied Mathematics, 1, 100002. https://doi.org/10.1016/j.padiff.2020.100002
15. Tamarkin, J. (1928). Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions. Mathematische Zeitschrift, 27, 1–54.