1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

Analysis and Assessment of the Effects of Corrosive Hydrogen Media on the Stress-Strain State of a Spherical Titanium Alloy Shell

We consider the creation of a mathematical model describing the effect of corrosive hydrogen environment on the stress state of a hollow spherical shell made of titanium alloy grade VT1-0, the load is evenly distributed throughout the shell. The solution of the problem in practice was carried out by two-step method of sequential perturbation of parameters using MatLab and Maple programs. To solve the system of solving differential equations the finite difference method was applied. The solution of the diffusion equation of the aggressive hydrogen medium has been considered and the obtained solution has been compared with the results of the classical theory which does not take into account the aggressive effect of the corrosive medium.

Numerical solutions to a fractional diffusion equation used in modelling dye-sensitized solar cells

Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement. doi:10.1017/S1446181121000353

Numerical analyses of embankment dams containing plastic concrete cut-off walls using finite difference method (A case study: The Karkheh Embankment Dam)

In this paper, numerical analyses have been performed on the Karkheh embankment dam with a clayey core and plastic concrete cut-off wall during construction, impounding, and permanent seepage stages. The dam has 127 meters height and is located in a high seismic hazard zone in Iran. Different stages of construction, water impounding, and steady state seepage were modelled and analyzed using the hyperbolic and Mohr-Coulomb models with the two dimensional finite difference method (FDM). So, nonlinear analyses were performed using FLAC 2D to investigate the settlements and the pore water pressure changes in different zones of the dam during above-mentioned stages and the results were compared to those of the other studies. The results show that at the end of the construction stage, the maximum settlement equal to 1.45m occurs inside the clay core at the height of 65m. Then, after impounding of the reservoir and steady state stage, the maximum magnitude of the horizontal deformations occurs in the downstream of the dam equal to 0.55m; however, these magnitudes reach to 0.17m at the crest of the dam. Moreover, it was shown that the maximum horizontal displacement of the plastic concrete cut-off wall has happened at the top of the wall in the clay core which is in a good agreement with the other studies’ result.