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A novel algorithm for singularly perturbed parabolic differential-difference equations

Imiru Takele Dabaa Department of Mathematics, Dilla University, Dilla, EthiopiaCorrespondence[email protected]

https://orcid.org/0000-0002-7822-9679 View further author information

Gemechis File Duressab Department of Mathematics, Jimma University, Jimma, Ethiopia

https://orcid.org/0000-0003-1889-4690 View further author information

Lishan Liuc Qufu Normal University, Jining, Shandong, ChinaView further author information

(Reviewing editor:)

Article: 2133211 | Received 19 Aug 2022, Accepted 30 Sep 2022, Published online: 04 Dec 2022

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https://doi.org/10.1080/27684830.2022.2133211
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Table 1. $E_{ε, δ, η}^{N, τ}, r_{ε, δ, η}^{N, τ}$ $E^{N, τ}$ , $r^{N, τ}$ $(E_{ε, δ, η}^{N, τ}) e x t p$ $(r_{ε, δ, η}^{N, τ}) e x t p$ , $(E^{N, τ}) e x t p$ and $(r^{N, τ}) e x t p$ for Example, 5.1 using method (3.19) and scheme(3.19) with Richardson extrapolation (4.10) with $δ = η = 0.5 \times ε$

Display Table

Table 2. $E_{ε, δ, η}^{N, τ}, r_{ε, δ, η}^{N, τ}$ $E^{N, τ}$ , $r^{N, τ}$ $(E_{ε, δ, η}^{N, τ}) e x t p$ $(r_{ε, δ, η}^{N, τ}) e x t p$ , $(E^{N, τ}) e x t p$ and $(r^{N, τ}) e x t p$ for Example, 5.2 using method (3.19) and scheme(3.19) with Richardson extrapolation (4.10) with $δ = η = 0.5 \times ε, M = N$

Display Table

Table 3. $E_{ε, δ, η}^{N, τ}, r_{ε, δ, η}^{N, τ}$ $E^{N, τ}$ , $r^{N, τ}$ $(E_{ε, δ, η}^{N, τ}) e x t p$ $(r_{ε, δ, η}^{N, τ}) e x t p$ , $(E^{N, τ}) e x t p$ and $(r^{N, τ}) e x t p$ for Example, 5.3 using method (3.19) and method (3.19) with Richardson extrapolation (4.10) with $T = 1.0, δ = η = 0.5 \times ε$

Display Table

Table 4. $(E_{ε, δ, η}^{N, τ}) e x t p$ $(r_{ε, δ, η}^{N, τ}) e x t p$ , $(E^{N, τ}) e x t p$ and $(r^{N, τ}) e x t p$ for Example, 5.1 using proposed method (3.19) with Richardson extrapolation (4.10) and results in (Mesfin M. Woldaregay & Duressa, Citation2019) and (Woldaregay & Duressa, Citation2020) with $T = 3, δ = 0.6 \times ε, η = 0.5 \times ε$

Display Table

Figure 1. The effect of $ε$ on the solution profiles at $δ = 0.6 \times ε$ and $η = 0.5 \times ε$ .

Figure 1. The effect of ε on the solution profiles at δ=0.6×ε and η=0.5×ε.

Figure 2. The effect of $δ$ on the solution profiles at $ε = 2^{- 4}, η = 0.5 \times ε$ .

Figure 2. The effect of δ on the solution profiles at ε=2−4,η=0.5×ε.

Figure 3. The effect of $η$ on the solution profiles with $ε = 2^{- 4}, δ = 0.6 \times ε$ .

Figure 3. The effect of η on the solution profiles with ε=2−4,δ=0.6×ε.

Figure 4. Numerical solution profiles for Example, 5.1 at $δ = 0.6 \times ε, η = 0.5 \times ε, N = M = 512$ .

Figure 4. Numerical solution profiles for Example, 5.1 at δ=0.6×ε,η=0.5×ε,N=M=512.

Figure 5. Numerical solution profiles for Example, (5.2) at $T = 2.0, δ = 0.6 \times ε, η = 0.5 \times ε$ and $N = M = 512$ .

Figure 5. Numerical solution profiles for Example, (5.2) at T=2.0,δ=0.6×ε,η=0.5×ε and N=M=512.

Figure 6. Numerical solution profiles for Example, 5.3 at $T = 1.0, δ = 0.6 \times ε, η = 0.5 \times ε$ and $N = M = 512$ .

Figure 6. Numerical solution profiles for Example, 5.3 at T=1.0,δ=0.6×ε,η=0.5×ε and N=M=512.

Figure 7. Log-log plot for the maximum absolute errors before and after the Richardson extrapolation technique.

Kumar, D. (2018). An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience. Numerical Methods for Partial Differential Equations, 34(6), 1933–1952. https://doi.org/10.1002/num.22269

Web of Science ®Google Scholar

Woldaregay, M. M., & Duressa, G. F. (2022). Uniformly convergent numerical method for singularly perturbed delay parabolic differential equations arising in computational neuroscience. Kragujevac Journal of mathematics, 46(1), 65–84.

Web of Science ®Google Scholar

Woldaregay, M. M., & Duressa, G. F. (2019). Parameter uniform numerical method for singularly perturbed parabolic differential difference equations. Journal of the Nigerian Mathematical Society, 38(2), 223–245. https://ojs.ictp.it/jnms/index.php/jnms/article/view/474

Google Scholar

Woldaregay, M. M., & Duressa, G. F. (2020). Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations. ITM Web of Conferences, 34, 02011. https://doi.org/10.1051/itmconf/20203402011

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A novel algorithm for singularly perturbed parabolic differential-difference equations

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Table 1. Eε,δ,ηN,τ,rε,δ,ηN,τ EN,τ, rN,τ (Eε,δ,ηN,τ)extp (rε,δ,ηN,τ)extp, (EN,τ)extp and (rN,τ)extp for Example, 5.1 using method (3.19) and scheme(3.19) with Richardson extrapolation (4.10) with δ=η=0.5×ε

Table 2. Eε,δ,ηN,τ,rε,δ,ηN,τ EN,τ, rN,τ (Eε,δ,ηN,τ)extp (rε,δ,ηN,τ)extp, (EN,τ)extp and (rN,τ)extp for Example, 5.2 using method (3.19) and scheme(3.19) with Richardson extrapolation (4.10) with δ=η=0.5×ε,M=N

Table 3. Eε,δ,ηN,τ,rε,δ,ηN,τ EN,τ, rN,τ (Eε,δ,ηN,τ)extp (rε,δ,ηN,τ)extp, (EN,τ)extp and (rN,τ)extp for Example, 5.3 using method (3.19) and method (3.19) with Richardson extrapolation (4.10) with T=1.0,δ=η=0.5×ε

Table 4. (Eε,δ,ηN,τ)extp (rε,δ,ηN,τ)extp, (EN,τ)extp and (rN,τ)extp for Example, 5.1 using proposed method (3.19) with Richardson extrapolation (4.10) and results in (Mesfin M. Woldaregay & Duressa, Citation2019) and (Woldaregay & Duressa, Citation2020) with T=3,δ=0.6×ε,η=0.5×ε

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