This study focuses on a nonlinear viscoelastic wave equation involving logarithmic nonlinearity. It considers a nonlinear distributed delay influencing the boundary feedback, which is coupled with acoustic and fractional boundary conditions. Following the proof of global existence, we demonstrate the exponential growth and blow-up of solutions with positive initial energy under appropriate assumptions and for a general case of the kernel. This finding broadens and enhances earlier results.
In this research, a machine learning method based on physics informed neural network and fractional-order Genocchi wavelets (FGWs) as activation function is explored to solve delay Hilfer fractional differential equations (DHFDEs). In this machine learning algorithm, the FGWs and sinh$$ \sinh $$ functions are used as kernel functions to approximate the solution of DHFDEs. In fact, the solution of DHFDEs is approximated as a combination of the mentioned kernel functions and a set of weights that are learned during the fitting process. We apply the roots of the Legendre functions as training data to develop the algorithm. Then, the training is proposed using the optimizer algorithm. In addition, the error bound of the presented strategy is discussed. Finally, to illustrate the validity and feasibility of our results, three numerical simulation along with several tables and figures are utilized.
In order to create biomaterials, this review investigates the extraordinary potential of white rot fungi (WRF) as natural engineering with the special ability to break down complex organic molecules, such as lignocellulosic biomass, WRF, in particular, have enzymes that are capable of breaking down lignin. By taking advantage of this capacity, mycelium-based biomaterials provide a sustainable substitute for traditional materials made from fossil fuels, assisting in the decrease of carbon emissions and the ameliorsation of environmental deterioration. Additionally, the discussion highlights the prospective uses of mycelium-based biomaterials including pure mycelium material and mycelium-based composite incorporating across various sectors for diverse applications, such as packaging, bio-leather, biobandages for wound healing, construction and mycoelectronics, with a focus on their biocompatibility, adaptability, and large-scale manufacturing potential through the explanation of the mutually beneficial interaction of WRF mycelium and biomaterial development, this paper highlights the critical role of WRF mycelium that plays in promoting long-term solutions related to environmental issues. Moreover, the biomaterials obtained from WRF has the potential to alter material science and advance the circular economy paradigm, which turns waste into useful resources and promotes a more sustainable and environmentally friendly future.
This study is concerned with a nonlinear viscoelastic wave equation. By supposing the nonlinear time-varying delay feedback acting on the boundary coupling by the acoustic and fractional boundary conditions. Under suitable assumptions and in the case general of the kernel, we prove the asymptotic behavior of solutions.
This paper presents an optimization algorithm designed to effectively handle a new general class of the nonlinear variable-order fractional partial differential equations (GCNV-OFPDEs) with nonlocal boundary conditions. Our approach involves utilizing a novel variant of the polynomials, namely generalized Abel polynomials (GAPs), and also new operational matrices to approximate the solution of the GCNV-OFPDEs. A key aspect of our algorithm is the transformation of GCNV-OFPDEs, along with their respective nonlocal boundary conditions, into systems of nonlinear algebraic equations. By solving these systems, we can determine the unknown coefficients and parameters. To address the nonlinear system, we employ the Lagrange multipliers to achieve optimal approximations. The convergence analysis of the approach is discussed. To validate the effectiveness of our algorithm, we conducted numerous experiments using various examples. The results obtained demonstrate the exceptional accuracy of our approach and its potential for extension to more complex problems in the future.
