## Obtaining exact analytical solutions for nonstationary heat conduction problems using orthogonal methods.(Russian. English summary)Zbl 1413.35238

Summary: Through the interplay of orthogonal methods by L. V. Kantorovich, Bubnov-Galerkin and a heat balance integral method there have been obtained an exact analytical solution of a nonstationary heat conduction problem for an infinite plate under the symmetrical first-type boundary conditions. It was possible to obtain an exact solution through the employment of approximate methods due to the appliance of trigonometric coordinate functions, possessing the property of orthogonality. They enable us to determine eigenvalues not through the solution of the Sturm-Liouville boundary value problem, which supposes the second-order differential equation integration, but through the solution of a differential equation for an unknown function on time, which is the first-order equation. Due to the property of coordinate functions mentioned above, while determining constants of integration out of initial conditions it is possible to avoid solving large systems of algebraic linear equations with ill-conditioned matrix of coefficients. Thus, it simplifies both the process of obtaining a solution and its final formula and provides an opportunity to find not only an approximate, but also an exact analytical solution, represented by an infinite series.

### MSC:

 35K05 Heat equation 80A20 Heat and mass transfer, heat flow (MSC2010) 35C10 Series solutions to PDEs
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### References:

   Kantorovich L. V., Krylov V. I., Priblizhennye metody vysshego analiza [Approximate Methods of Higher Analysis], Fizmatgiz, Leningrad, 1962, 708 pp. (In Russian) · Zbl 0040.21503   Tsoi P. V., Sistemnye metody rascheta kraevykh zadach teplomassoperenosa [System methods for calculating the heat and mass transfer boundary value problems], Moscow Power Engineering Institute Publ., Moscow, 2005, 568 pp. (In Russian)   Kudinov V. A., Kartashov E. M., Kalashnikov V. V., Analiticheskie resheniya zadach teplomassoperenosa i termouprugosti dlya mnogosloynykh konstruktsiy [Analytical problem solving heat and mass transfer and thermoelasticity for multilayer structures], Vysshaya shkola, Moscow, 2005, 430 pp. (In Russian)   Kudinov V. A., Averin B. V., Stefaniuk E. V., Teploprovodnost’ i termouprugost’ v mnogosloinykh konstruktsiiakh [Thermal Conductivity and Thermoelasticity in Multilayer Structures], Vysshaia shkola, Moscow, 2008, 305 pp. (In Russian)   Lykov A. V., Teoriia teploprovodnosti [Theory of heat conduction], Vysshaia shkola, Moscow, 1967, 600 pp. (In Russian)
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