Approximate Solutions to Time-fractional Models by Integral-balance Approach

What is it about?

The chapter is an attempt to collate results on approximate integral-balance solutions of evolution equations involving fractional-time derivatives in the sense of Riemann-Liouville or Caputo. The examples encompass a subdiusion equation of arbitrary order and transient ows of viscoelastic uids. The approximate solutions are in closed form and allow clearly see the retarding eect of the fractional order (aecting the power-law memory kernel in the fractional derivative) on the development of the diusion processes

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Why is it important?

This chapter was written with a single idea of collating results on the integral-balance approach applied to models involving time-fractional derivatives. This analysis could be considered as an attempt to strike the balance between a series of articles utilizing the method, and dissemination of the main achievements, as well as to show emerging problems. The examples included in the chapter are solved step-by step and demonstrate how this should be done and what the accuracy of approximation is. The main problems are models with Riemann-Liouville derivative, but the case of subdiusion equation clearly demonstrates that the method works with Caputo derivatives, too. In addition to the rst results developed by the simple Heat-balance integral (HBIM), this chapter collates recently developed results obtained by the Double- Integration method (DIM), working very eective with both Riemann-Liouville and Caputo derivatives. The chapter evaluates some specic features of the integration technique termed Frozen-Front approach (FFA) used together with HBIM in the rst publication on the Approximate Solutions to Time-fractional Models Ë 107 integral-balance models on solutions of models with time-fractional derivatives. As a general outcome, FFA is practically equivalent to HBIM and in many cases the solutions of these integration techniques coincide. All the solutions were performed by a general parabolic prole. This prole gives more freedom in the tuning of the approximate solution through search of the optimal exponent in contrast to the case when polynomial proles (quadratic or cubic) can be used. In the latter cases, the order of the polynomial prole predetermines the accuracy of the approximation. There are enough comments accompanying the solutions of the examples solved in this chapter, so it would be better to end the discussion and to give a freedom to the readers to estimate what is really the power of the method and what would be further developed on its basis. I am grateful to all

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