The closed-form solutions for a model with technology diffusion via Lie symmetries

Azam Chaudhry; Rehana Naz

doi:10.3934/dcdss.2024133

Article Contents

2025, Volume 18, Issue 4: 1036-1053. Doi: 10.3934/dcdss.2024133

This issue Previous Article Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power Next Article Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several invariant solutions to a (2+1)-dimensional Hirota bilinear equation

The closed-form solutions for a model with technology diffusion via Lie symmetries

Azam Chaudhry^1, , and
Rehana Naz^2,

1.
Department of Economics, Lahore School of Economics, Lahore, 53200, Pakistan
2.
Department of Mathematics, Lahore School of Economics, Lahore, 53200, Pakistan

^*Corresponding author: Azam Chaudhry
^*Corresponding author: Azam Chaudhry

Received: February 2024

Revised: July 2024

Early access: August 2024

Published online: August 2024

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Abstract

A critical component of economic growth is growth in productivity which is dependent on technology adoption. While most technologies are created in developed economies, they diffuse to developing economies through various channels such as trade, migration and knowledge spillovers. In this paper, we develop the first model that integrates compartmental models with diffusion to analyze technology adoption within a framework of a system of second-order non-linear partial differential equations. We employ Lie point symmetries to derive reductions and closed-form solutions for a model of technology diffusion. A three-dimensional Lie algebra is established for the model. We utilize the combinations of Lie symmetries to obtain reductions and establish closed-form solution for the technology diffusion model. Additionally, we utilise the closed-form solution to provide graphical representations of the technology diffusion process over effective distance and over time and find the commonly observed S-curve path of technology diffusion. Furthermore, we conduct a sensitivity analysis to develop policy insights into the factors influencing the diffusion of technology.

Keywords:

Mathematics Subject Classification: Primary: 35B06, 35C05, 58D19, 91-10; Secondary: 35Q91, 91B02.

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Figure 1. The graph of initial distribution of technology-receptor and technology-adopter countries across the domain for different values of $ \nu $ when $ 0\leq x\leq 10 $

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Figure 2. Surface plots of $ T_R(t, x) $ and $ T_A(t, x) $ with parameters $ \nu = 0.06 $, $ \beta = 0.4 $, $ \tau = 0.3 $, $ d_R = 0.002 $, $ d_A = 0.003 $, $ R_0 = 99 $, $ A_0 = 1 $ over time interval $ 0\leq t\leq 150 $ and spatial range $ 0\leq x\leq 10 $

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Figure 3. Graphs of $ T_R(t) $ and $ T_A(t) $ with parameters $ \nu = 0.06 $, $ \beta = 0.4 $, $ \tau = 0.3 $, $ d_R = 0.002 $, $ d_A = 0.003 $, $ R_0 = 99 $, $ A_0 = 1 $ over time interval $ 0\leq t\leq 150 $ for fixed values of $ x $

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Figure 4. Graphs of $ T_R(x) $ and $ T_A(x) $ with parameters $ \nu = 0.06 $, $ \beta = 0.4 $, $ \tau = 0.3 $, $ d_R = 0.002 $, $ d_A = 0.003 $, $ R_0 = 99 $, $ A_0 = 1 $ over the spatial range $ 0\leq x\leq 10 $ for fixed values of $ t $

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Figure 5. Effect of change of $ \beta $ on $ T_R(t) $, $ T_A(t) $ and $ \Omega(t) $ with other parameters as $ \nu = 0.06 $, $ \tau = 0.3 $, $ d_R = 0.002 $, $ d_A = 0.003 $, $ R_0 = 99 $, $ A_0 = 1 $ over time interval $ 0\leq t\leq 150 $ for fixed value of $ x = 4 $

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Figure 6. Effect of change of $ \tau $ on $ T_R(t) $, $ T_A(t) $ and $ \Omega(t) $ with other parameters as $ \nu = 0.06 $, $ \beta = 0.4 $, $ d_R = 0.002 $, $ d_A = 0.003 $, $ R_0 = 99 $, $ A_0 = 1 $ over time interval $ 0\leq t\leq 150 $ for fixed value of $ x = 4 $

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