The aim of the research is design efficient neural networks to solve important type of partial differential equations represent many important models. The efficiency of design based on chooses a suitable activation function, suitable training algorithms with study how to increase the speed of this algorithm. Then we used suggested design to solve third order PDEs.

This article presents the mathematical model of unsteady state horizontal radial flow in homogenous and anisotropic confined aquifers in polar coordinate which represented an important problem. We design optimal neural network to solve this problem. Also, we estimate the parameters of transmissivity aquifer with high accuracy. The optimality depending on choosing optimum architectural with suitable training algorithm and transfer function. The best design neural network is trained by new training algorithm say LNA. The retrained almost fast by suitable transferring function. Based on the modified architecture, the training and testing phases of the solving process are divided into two. The dataset is separated into three sections during the training phase: 60% of the training data, 20% of the validation data, and 20% of the test data. The total square error (TSE) for the trained phase with using the bach propagation algorithm (BPA) is 4.1499e+01 while it is 9.5898e-03 if we using suggested training algorithm (LNA). The LNA has a much smaller TSE when compared to the BPA. These much smaller values of the TSE indicate that LNA performs better than the BPA for the same number of iterations. So suggested architecture has many advantages such loss function computed on a random sample of the domain, high performance, avoid local minima and can be adapted for the online dynamic modeling, automation, control and robotics applications.

We investigated the Hankel determinant H 3 (1) for functionals in the analytic function class and obtained the sharp bound for the third Hankel determinant | e 3 e 5 − e 4 2 | for a subclass of analytic functions in this paper.

The authors introduced and addressed  several new subclasses  of the family of meromorphically multivalent -star-like functions in the punctured unit disk  in this study, which makes use of several higher order -derivatives. Many fascinating properties and characteristics are extracted systematically for each of these newly identified function classes. Distortion theorems and radius problems are among these characteristics and functions. A number of coefficient inequalities for functions belonging to the subclasses are studied, and discussed, as well as a suitable condition for them is set. The numerous results are presented in this study and the previous works on this subject are also connected together in this study.

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