Chebyshev collocation method for the variable-order fractional diffusion equation with a variable diffusion coefficient

In this paper, we study the space-time variable-order fractional diffusion equation with a variable diffusion coefficient. The fractional derivatives of variable-orders are considered in the Caputo sense. We propose a numerically efficient pseudospectral method with Chebyshev polynomial as an orthogonal basis function. Also, we examine the error analysis of the given numerical approach. A variation on the maximum absolute error with the different variable orders in space and time are studied. Some illustrative examples are presented with different boundary conditions, e.g., Dirichlet, mixed, and non-local. The applicability of the method is also tested with the problem that has fractional power in solution. The results obtained from the proposed method prove the efficacy and reliability of the method.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

Multi-term time fractional diffusion equations and novel parameter estimation techniques for chloride ions sub-diffusion in reinforced concrete

In this paper, searching for a better chloride ions sub-diffusion system, a multi-term time-fractional derivative diffusion model is proposed for the description of the time-dependent chloride ions penetration in reinforced concrete structures exposed to chloride environments. We prove the stability and convergence of the model. We use the modified grid approximation method (MGAM) to estimate the fractional orders and chloride ions diffusion coefficients in the reinforced concrete for the multi-term time fractional diffusion system. And then to verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, two numerical examples with real data are investigated. Meanwhile, we use two methods of fixed chloride ions diffusion coefficient and variable diffusion coefficient with diffusion depth to simulate chloride ions sub-diffusion system. The result shows that with the new fractional orders and parameters, our multi-term fractional order chloride ions sub-diffusion system is capable of providing numerical results that agree better with the real data than other models. On the other hand, it is also noticed from the numerical solution of the chloride ions sub-diffusion system that setting the variable diffusion coefficient with diffusion depth is more reasonable. And it is also found that chloride ions diffusion coefficients in reinforced concrete should be decreased with diffusion depth which is completely consistent with the theory. In addition, the model can be used to predict the chloride profiles with a time-dependent property. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.