Solutions of Bagley–Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm with error estimates

Adaptive Technique for Solving 1-D Interface Problems of Fractional Order

In this paper, we present a numerical technique for solving 1-D interface problems of fractional order. This technique relies on the reproducing kernel functions and the shooting method. The biggest advantage over the existing standard analytical techniques is overcoming the difficulty arising in calculating complicated terms. Numerical examples are inspected to feature the significant highlights of this technique. Moreover, the solution procedure is simple, more effective and clearer.

Prediction of hydraulic blockage at culverts from a single image using deep learning

Cross-drainage hydraulic structures such as culverts and bridges in urban landscapes are prone to get blocked by the transported debris (e.g., urban, vegetated), which often reduces their hydraulic capacity and triggers flash floods. Unavailability of relevant data from blockage-originated flooding events and complex nature of debris accumulation are highlighted factors hindering the research within the blockage management domain. Wollongong City Council (WCC) blockage conduit policy is the leading formal guidelines to incorporate blockage into design guidelines; however, are criticized by the hydraulic engineers for its dependence on the post-flood visual inspections (i.e., visual blockage) instead of peak floods hydraulic investigations (i.e., hydraulic blockage). Apparently, no quantifiable relationship is reported between the visual blockage and hydraulic blockage; therefore, many consider WCC blockage guidelines invalid. This paper exploits the power of Artificial Intelligence (AI), motivated by its recent success, and attempts to relate visual blockage with hydraulic blockage by proposing a deep learning pipeline to predict hydraulic blockage from an image of the culvert. Two experiments are performed where the conventional pipeline and end-to-end learning approaches are implemented and compared in the context of predicting hydraulic blockage from a single image. In experiment one, the conventional deep learning pipeline approach (i.e., feature extraction using CNN and regression using ANN) is adopted. In contrast, in experiment two, end-to-end deep learning models (i.e., E2E_ MobileNet, E2E_ BlockageNet) are trained and compared with the conventional pipeline approach. Dataset (i.e., Hydraulics-Lab Blockage Dataset (HBD), Visual Hydraulics-Lab Dataset (VHD)) used in this research were collected from laboratory experiments performed using scaled physical models of culverts. E2E_ BlockageNet model was reported best in predicting hydraulic blockage with $$R^2$$ R 2 score of 0.91 and indicated that hydraulic blockage could be interrelated with the visual features at the culvert.

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley–Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.