Solving a Sequence of Recurrence Relations for First-Order Differential Equations

Andrey Batukhtin; Irina Batukhtina; Maxim Bass; Sergey Batukhtin; Pavel Safronov; Marina Baranovskaya

doi:10.12973/ejmste/79043

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Andrey Batukhtin ¹ ^* , Irina Batukhtina ¹, Maxim Bass ¹, Sergey Batukhtin ¹, Pavel Safronov ¹, Marina Baranovskaya ¹

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¹ Transbaikal State University, RUSSIA
^* Corresponding Author

Abstract

A whole category of engineering and economic problems can be reduced to solving a set of differential equations. Downsides of known approaches for their solutions include limited accuracy numerical methods with stringent requirements for computational power. A direct analytical solution should be derived to eliminate such flaws. This research intends to derive such a solution for an n-dimensional set of recurrence relations for first-order differential equations, linearly dependent on the right side. The research methodology relies on successive integration of the considered set in view of the initial conditions. The overall solution was derived as a sum of products of exponential multipliers with constant coefficients that are defined through weights of a tree graph, which is a descriptor of successive integration. An analytical solution for an n-dimensional set of recurrent differential equations in view of the initial conditions has been derived for the first time in this research.

Keywords

analytical condition
differential equations
differentiation
initial conditions

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

https://doi.org/10.12973/ejmste/79043

EURASIA J Math Sci Tech Ed, 2017 - Volume 13 Issue 11, pp. 7179-7191

Publication date: 24 Oct 2017

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Article Downloads: 1347

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References

Bau, U., Braatz, A. L., Lanzerath, F., Herty, M., & Bardow, A. (2015). Control of adsorption chillers by a gradient descent method for optimal cycle time allocation. International Journal of Refrigeration-Revue Internationale du Froid, 56, 52-64. doi:10.1016/j.ijrefrig.2015.03.026.
Bég, O. A., Ali, N., & Zaman, A. et al. (2016). Computational modeling of heat transfer in an annular porous medium solar energy absorber with the P1-radiative differential. Journal of the Taiwan Institute of Chemical Engineers, 66, 258-268. doi:10.1016/j.jtice.2016.06.034.
Bolatturk, A. (2006). Determination of optimum insulation thickness for building walls with respect to various fuels and climate zones in Turkey. Appl Therm Eng, 26(11-12), 1301–9. doi:10.1016/j.applthermaleng.2005.10.019.
Etter, D. M., Kuncicky, D., & Moore, H. (2004). Introduction to MATLAB 7. Springer.
Gao, H., Chui, E., & Runstedler, A. et al. (2010). Numerical investigation of plasma ignition process in a utility boiler. Proceedings of 6th International Workshop and Exhibition on Plasma Assisted Combustion (IWEPAC). Heilbronn, Germany, 13-15 September, 69.
Gholampour, M., & Ameri, M. (2015). Design Considerations of Photovoltaic/Thermal Air Systems: Energetic and Exergetic Approaches. Journal of Solar Energy Engineering. Transactions of the ASME, 137(3). Article: 031005. doi:10.1115/1.4029107.
Goryachikh, N. V., Batukhtin, A. G., & Ivanov, S. A. (2010). Some Methods for Making Cogeneration Stations More Maneuverable. Thermal Engineering, 57(10), 892–896.
Hussain, M. I., Ali, A., & Lee, G. H. (2016). Multi-module concentrated photovoltaic thermal system feasibility for greenhouse heating: Model validation and techno-economic analysis. Solar Energy, 135, 719-730. doi:10.1016/j.solener.2016.06.053.
Kalema, T., Jóhannesson, G., Pylsy, P., & Hagengran, P. (2008). Accuracy of Energy Analysis of Buildings: A Comparison of a Monthly Energy Balance Method and Simulation Methods in Calculating the Energy Consumption and the Effect of Thermal Mass. Journal of Building Physics, 32(2), 101-130. doi:10.1177/1744259108093920.
Kaminski, K., & Krzyzynski, T. (2016). Modeling and Simulation of the Solar Collector Using Different Approaches. Mechatronics: Ideas, Challenges, Solutions and Applications, 414, 131-151. doi:10.1007/978-3-319-26886-6_9.
Karpenko, E. I., Rinchinov, A. P., Karpenko, Y. E., Bass, M. S., & Batukhtin, S. G. (2016). The results of the tests of experimental-industrial plasma-cyclone installation. Promyshlennaya Energetika, 4, 24-27.
Karpenko, E. I., & Trusov, B. G. (1995). А Comparative Analysis of Plasma and Fire Technologies of Pulverized Coal Ignition, Combustion and Gasification Using a Mathematical Model of Chemically Non-equilibrium System. Thermophysics and Aeromechanics, 2(23), 245-250.
Khelifa, A., Touafek, K., & Ben M. (2015). Approach for the modelling of hybrid photovoltaic-thermal solar collector. IET Renewable Power Generation, 9(3), 207-217. doi:10.1049/iet-rpg.2014.0076.
Khelifa, A., Touafek, K., Ben Moussa, H., & Tabet, I. (2016). Modeling and detailed study of hybrid photovoltaic thermal (PV/T) solar collector. Solar Energy, 135, 169-176. doi:10.1016/j.solener.2016.05.048.
Kicsiny, R. (2017). Grey-box model for pipe temperature based on linear regression. International Journal of Heat and Mass Transfer, 107, 13-20. doi:10.1016/j.ijheatmasstransfer.2016.11.033.
Kicsiny, R. (2016). Improved multiple linear regression based models for solar collectors. Renewable Energy, 91, 224–232. doi:10.1016/j.renene.2016.01.056.
Kicsiny, R. (2014). New delay differential equation models for heating systems with pipes. Int. J. Heat Mass Transfer, 79, 807–815. doi:10.1016/j.ijheatmasstransfer.2014.08.058.
Li, H., & Chen, Z. (2008). Overview of different wind generator systems and their comparisons. IET Renewable Power Generation, 2(2), 123-138. doi:10.1049/iet-rpg:20070044.
Messerle, V. E., Karpenko, E. I., & Ustimenko, A. B. (2014). Plasma Assisted Power Coal Combustion in the Furnace of Utility Boiler: Numerical Modelling and Full-Scale Test. Fuel, 126, 294-300. doi:10.1016/j.fuel.2014.02.047.
Messerle, V. E., Karpenko, E. I., Ustimenko, A. B., & Lavrichshev, O. A. (2013). Plasma preparation of coal to combustion in power boilers. Fuel Processing Technology, 107, 93-98. doi:10.1016/j.fuproc.2012.07.001.
Messerle, V. E., Mosse, A. L., & Ustimenko, A. B. (2016). Municipal Solid Waste Plasma Processing: Thermodynamic Computation and Experiment. IEEE Transactions on Plasma Science, 99, 1-6. doi:10.1109/TPS.2016.2601107.
Messerle, V. E., Ustimenko, A. B., & Lavrichshev, O. A. (2016), Comparative study of coal plasma gasification: Simulation and experiment. Fuel, 164, 172-179. doi:10.1016/j.fuel.2015.09.095.
Polyanin, A. D., & Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton.
Romero, J. C., & Linares, P. (2014). Exergy as a global energy sustainability indicator. A review of the state of the art. Renewable and Sustainable Energy Reviews, 33, 427-442. doi:10.1016/j.rser.2014.02.012.
Saleh, A. M., Mueller, D. W., & Abu-Mulaweh, H. I. (2015). Flat-Plate Solar Collector in Transient Operation: Modeling and Measurements. Journal of Thermal Science and Engineering Applications, 7. doi:10.1115/1.4028569.
Sheng, S., & Tu, Z. C. (2013). Universality of energy conversion efficiency for optimal tight-coupling heat engines and refrigerators. Journal of Physics A: Mathematical and Theoretical, 46(40), 402001. doi:10.1088/1751-8113/46/40/402001.
Smith, E., Koolnapadol, N., & Promvonge, P. (2012). Heat transfer behavior in a square duct with tandem wire coil element insert. Chin. J. Chem. Eng., 20, 863–869. wos:000311472200008.
Stepanov, V., Starikova, N., & Stepanova, T. (2000). Indices for estimation of energy conservation in space heating. Energy and Buildings, 31(3), 189-193. doi:10.1016/S0378-7788(99)00013-4.
Stevanovic, V. D., Zivkovic, B., Prica, S., Maslovaric, B., Karamarkovic, V., & Trkulja, V. (2009). Prediction of thermal transients in district heating systems. Energy Convers. Manage, 50, 2167-2173. doi:10.1016/j.enconman.2009.04.034.
Thianpong, C., Yongsiri, K., Nanan, K., & Eiamsa-ard, S. (2012). Thermal performance evaluation of heat exchangers fitted with twisted-ring turbulators. Int. Commun. Heat Mass Transf., 39, 861-868. doi:10.1016/j.icheatmasstransfer.2012.04.004.
Torío, H., Angelotti, A., & Schmidt D. (2009). Exergy analysis of renewable energy-based climatisation systems for buildings: a critical view. Energy and Buildings, 41(3), 248-271.
Valero, A. (2006). Exergy accounting: capabilities and drawbacks. Energy, 31(1), 164-180. doi:10.1016/j.energy.2004.04.054.

How to cite this article

APA

Batukhtin, A., Batukhtina, I., Bass, M., Batukhtin, S., Safronov, P., & Baranovskaya, M. (2017). Solving a Sequence of Recurrence Relations for First-Order Differential Equations. Eurasia Journal of Mathematics, Science and Technology Education, 13(11), 7179-7191. https://doi.org/10.12973/ejmste/79043

Vancouver

Batukhtin A, Batukhtina I, Bass M, Batukhtin S, Safronov P, Baranovskaya M. Solving a Sequence of Recurrence Relations for First-Order Differential Equations. EURASIA J Math Sci Tech Ed. 2017;13(11):7179-91. https://doi.org/10.12973/ejmste/79043

AMA

Batukhtin A, Batukhtina I, Bass M, Batukhtin S, Safronov P, Baranovskaya M. Solving a Sequence of Recurrence Relations for First-Order Differential Equations. EURASIA J Math Sci Tech Ed. 2017;13(11), 7179-7191. https://doi.org/10.12973/ejmste/79043

Chicago

Batukhtin, Andrey, Irina Batukhtina, Maxim Bass, Sergey Batukhtin, Pavel Safronov, and Marina Baranovskaya. "Solving a Sequence of Recurrence Relations for First-Order Differential Equations". Eurasia Journal of Mathematics, Science and Technology Education 2017 13 no. 11 (2017): 7179-7191. https://doi.org/10.12973/ejmste/79043

Harvard

Batukhtin, A., Batukhtina, I., Bass, M., Batukhtin, S., Safronov, P., and Baranovskaya, M. (2017). Solving a Sequence of Recurrence Relations for First-Order Differential Equations. Eurasia Journal of Mathematics, Science and Technology Education, 13(11), pp. 7179-7191. https://doi.org/10.12973/ejmste/79043

MLA

Batukhtin, Andrey et al. "Solving a Sequence of Recurrence Relations for First-Order Differential Equations". Eurasia Journal of Mathematics, Science and Technology Education, vol. 13, no. 11, 2017, pp. 7179-7191. https://doi.org/10.12973/ejmste/79043