The method of solving problems of mathematical physics, in particular for pressure distribution finding in the water in the underground gas storage layers on the basis of the biorthogonal polynomials constructed by the authors is proposed in the paper. The way of the problem solving by the method of separation of variables on the basis of the biorthogonal polynomials is studied. The solution of the problem is found in the form of the series sum of the biorthogonal and quasi-spectral polynomials. The comparative analysis for the different values of parameters is performed. The impact of the methods parameters, in particular the partial sum order, the bit grid and the calculation error on the accuracy of the solution obtained is studied. The calculation results are presented in the form of the tables. The algorithm of the process of the gas motion in the pipelines using fractional derivatives is constructed.

- Prytula N. M., Pyanylo Ya. D., Prytula M. H. The underground gas storages (mathematical models and methods). Lviv, RASTR-7 (2015), (in Ukrainian).
- Pyanylo Ya. D. Projection-iterative methods of solving of direct and inverse problems of transport. Lviv, Spline (2011), (in Ukrainian).
- Pyanylo Ya. D., Vavrychuk P. H. Determination of the motion velocity of gas-water contact at the process of the underground gas storages work. Physical-Mathematical Modeling and Informational Technologies.
**18**, 165--172 (2013), (in Ukrainian). - Lopuh N. B., Pyanylo Ya. D. Numerical model of gas filtration in porous media using fractional time derivatives. Mathematical methods in Chemistry and Biology.
**2**(1), 98--104 (2014), (in Ukrainian). - Pyanylo Ya. D., Sobko V. H. Constructing and researching of biorthogonal polynomials on Chebyshev polynomials basis. Applied problems of machanics and mathematics.
**11**, 135--141 (2013), (in Ukrainian). - Pyanylo Ya., Sobko V. Researching of the spectral expansions properties at bases of orthogonal, quasi-orthogonal and biorthogonal polynomials. Physical-Mathematical Modeling and Informational Tecnologies.
**19**, 146--156 (2014), (in Ukrainian). - Ahmad B., Sivasundaram S. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems.
**4**(1), 134--141 (2010). - Ahmad B., Sivasundaram S. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput.
**217**(2), 480--487 (2010). - Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006).
- Podlubny I. Fractional Differential Equations. Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto (1999).
- Samko S. G., Kilbas A. A., Marichev O. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993).
- Vasiliev V. V., Simak L. A. Fractional calculus and approximation methods in the modeling of dynamic systems. Kiev, Scientific publication of NAS of Ukraine (2008), (in Russian).
- Ditkin V. A., Prudnikov A. P. Operational calculus. Moscow, High school (1975), (in Russian).
- Pyanylo Ya., Vasyunyk M., Vasyunyk I. Investigation of the spectral method of solving of fractional time derivatives in Laguerre polynomials basis. Physical-Mathematical Modeling and Informational Technologies.
**18**, 173--179 (2013), (in Ukrainian). - Pyanylo Ya. Use of fractional derivatives for analysis of nonstationary gas motion in pipelines in the presense of compressor stations and outlets. Physical-Mathematical Modeling and Informational Technologies.
**16**, 122--132 (2012), (in Ukrainian).