Methods of Mathematical Modelling by Singh Harendra; Kumar Devendra; Baleanu Dumitru

Author:Singh, Harendra; Kumar, Devendra; Baleanu, Dumitru
Language: eng
Format: epub
Publisher: Taylor & Francis Group
Published: 2020-07-15T00:00:00+00:00

Let us now analyze the effect of the order ρ in the diffusion process when we stipulate the exponent n = 3.

In Figures 6.3 and 6.4, we plot with t = 0.9, the profile of the approximate solution of the fractional diffusion equation. We fix the order α = 1. The values of the order ρ increase and are less than 1. Clearly, all the curves decay rapidly and converge to zero. Furthermore, the order of the profiles from left to right follows a decrease in the order ρ. We notice an acceleration effect. The order ρ of the Caputo-generalized fractional derivative has an acceleration effect in the diffusion process.

In Figure 6.5, we plot with t = 0.9 the profile of the approximate solution of the generalized fractional diffusion equation. We fix the order α = 1. The values of the order ρ increase and is up to 1. We notice that all the curves decay. The order of the profiles from right to left follows an increase in the order ρ. We notice a retardation effect. The order ρ of the Caputo-generalized fractional derivative has a retardation effect in the diffusion process.

FIGURE 6.3

FIGURE 6.4

HBIM ρ < 1, n = 3.

FIGURE 6.5

HBIM ρ > 1, n = 3.

Let us now express the solution of an approximate solution of the fractional diffusion equation described by the Caputo-generalized fractional derivative when the penetration depth is got by the DIM.

Approach with the DIM: We stipulate the exponent n = 3. We obtain the following approximate solution of the generalized fractional diffusion equation

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