Artificial ANNs (ANNs) are relatively new computational tools used in the development of intelligent systems, some of which are inspired by biological ANNs, and have found widespread application in the solving of a variety of complex real-world problems. It boasts enticing features as well as remarkable data processing capabilities. In this paper, a comprehensive overview of the backpropagation algorithm for digital image denoising was discussed. Then, we presented a probabilistic analysis of how different algorithms address this challenge, arguing that small ANNs can denoise small-scale texture patterns almost as effectively as their larger equivalents. The results also show that self-similarity and ANNs are complementary paradigms for patch denoising, as demonstrated by an algorithm that effectively complements BM3D with small ANNs, surpassing BM3D at a low cost. Here, one of the most significant advantages of this learning technique is that, once taught, digital images may be recovered without prior knowledge of the degradation model (noise/blurring) that caused the digital image to become distorted.

In this paper, we investigate the multiplicity results of some positive solutions for a system of Hadamard fractional differential equations with parameters and p-Laplacian operator subject to three-point boundary conditions which contains fractional derivatives. The proofs of our main result, multiplicity of positive solutions, are derived in terms of different values of parameters by using Guo–Krasnosel’skii’s fixed point theorem.

This article introduces a new iterative transform method and homotopy perturbation transform method along with a natural transform to analyze the multi-dimensional Navier-Stokes equations. To solve the fractional-derivative, the Caputo-Fabrizio definition of the fractional derivative was employed. Four examples were considered to examine the efficacy and accuracy of the proposed methods. The efficiency and accuracy were also demonstrated by the solution comparison via graphs. The proposed methods' convergence and uniqueness are also discussed. The methods mentioned above are straightforward and support a high rate of convergence.