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An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov–Shishkin and Vulanović–Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically with the help of software Mathcad. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time.

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically with the help of software Mathcad. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time. [ABSTRACT FROM AUTHOR]

Abstract: We consider the reconstruction of complex obstacles from few far-field acoustic measurements. The complex obstacle is characterized by its shape and an impedance function distributed along its boundary through Robin type boundary conditions. This is done by minimizing an objective functional, which is the distance between the given far-field information and the far-field of the scattered wave corresponding to the computed shape and impedance function. We design an algorithm to update the shape and the impedance function alternatively along the descent direction of the objective functional. The derivative with respect to the shape or the impedance function involves solving the original Helmholtz problem and the corresponding adjoint problem, where boundary integral methods are used. Further we implement level set methods to update the shape of the obstacle. To combine level set methods and boundary integral methods we perform a parametrization step for a newly updated level set function. In addition since the computed shape derivative is defined only on the boundary of the obstacle, we extend the shape derivative to the whole domain by a linear transport equation. Finally, we demonstrate by numerical experiments that our algorithm reconstruct both shapes and impedance functions quite accurately for non-convex shape obstacles and constant or non-constant impedance functions. The algorithm is also shown to be robust with respect to the initial guess of the shape, the initial guess of the impedance function and even large percentage of noise. [Copyright &y& Elsevier]