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Fractional diffusion partial differential equation (PDE) models are used to describe anomalous transport phenomena in fractal porous media, where traditional diffusion models may not be applicable due to the presence of long-range dependencies and non-local behaviors. This study presents an efficient hybrid meshless method to the compute numerical solution of a two-dimensional multiterm time-fractional convection-diffusion equation. The proposed meshless method employs multiquadric-cubic radial basis functions for the spatial derivatives, and the Liouville-Caputo derivative technique is used for the time derivative portion of the model equation. The accuracy of the method is evaluated using error norms, and a comparison is made with the exact solution. The numerical results demonstrate that the suggested approach achieves better accuracy and computationally efficient performance.

Resonant optical cavities are essential components in mid-infrared applications. However, typical film-type cavities require multilayer stacks with a micron-thick spacer due to mid-infrared wavelengths, and their performance is limited by narrow frequency tunability and angular sensitivity. We propose and experimentally demonstrate the subwavelength-scale (≈λ0/150) resonant nanocavity arrays that enhance the absorption spectrum of the device in the mid-infrared (10–12 microns) via excitation of coupled surface plasmon–phonon polaritons. The proposed metal–insulator–polar dielectric (gold–silicon–silicon carbide) structure supports a guided mode of the coupled surface polaritons in the lateral direction while vertically confining the mid-infrared wave within the 80 nm thick dielectric spacer. In particular, the subwavelength-scale (≈λ0/10) gratings are imposed to form Fabry–Pérot cavity arrays displaying angle-insensitive and frequency-tunable absorption of up to 80% of the optical power in the mid-infrared. Our work should benefit diverse mid-infrared applications and novel designs of polariton-based photonic devices.

In this article, a hybrid Haar wavelet collocation method (HWCM) is proposed for the ill-posed inverse problem with unknown source control parameters. Applying numerical techniques to such problems is a challenging task due to the presence of nonlinear terms, unknown control parameter sources along the solution inside the given region. To find the numerical solution, derivatives are discretized adopting implicit finite-difference scheme and Haar wavelets. The computational stability and theoretical rate of convergence of the proposed HWCM are discussed in detail. Two numerical experiments are incorporated to show the well-condition behavior of the matrix obtained from HWCM and hence not required to supplement some regularization procedures. Moreover, the numerical solutions of the considered experiments illustrate the reliability, suitability, and correctness of HWCM.

In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.

Multi-term time-fractional partial differential equations (PDEs) have become a hot topic in the field of mathematical physics and are used to improve the modeling accuracy in the description of anomalous diffusion processes compared to the single-term PDE results. This research includes the numerical solutions of two-term time-fractional PDE models using an efficient and accurate local meshless method. Due to the advantages of the meshless nature and ease of applicability in higher dimensions, the demand for meshless techniques is increasing. This approach approximates the solution on a uniform or scattered set of nodes, resulting in well-conditioned and sparse coefficient matrices. Numerical tests are performed to demonstrate the efficacy and accuracy of the proposed local meshless technique.

In this study, the Daftardar–Jeffery polynomials are incorporated in the homotopy of the optimal homotopy asymptotic method (OHAM) for solving the generalized Hirota–Satsuma coupled system of Korteweg–de Vries equations. The results are displayed through graphs and tables. The results obtained in this study are also compared with the published work on OHAM, which shows that OHAM-DJ is more explicit, reliable, and an efficient analytical technique. The exactness of the developed method can be improved by performing the higher iterations.