In this paper, a new integral transform operator, which is similar to Fourier transform, is proposed for the first time. As a testing example, an application to the one-dimensional heat-diffusion ...problem is discussed. The result demonstrates accuracy and efficiency of the present technology to find the analytical solution for the heat-transfer problem.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The application of deep neural networks towards solving problems in science and engineering has demonstrated encouraging results with the recent formulation of physics-informed neural networks ...(PINNs). Through the development of refined machine learning techniques, the high computational cost of obtaining numerical solutions for partial differential equations governing complicated physical systems can be mitigated. However, solutions are not guaranteed to be unique, and are subject to uncertainty caused by the choice of network model parameters. For critical systems with significant consequences for errors, assessing and quantifying this model uncertainty is essential. In this paper, an application of PINN for laser bio-effects with limited training data is provided for uncertainty quantification analysis. Additionally, an efficacy study is performed to investigate the impact of the relative weights of the loss components of the PINN and how the uncertainty in the predictions depends on these weights. Network ensembles are constructed to empirically investigate the diversity of solutions across an extensive sweep of hyper-parameters to determine the model that consistently reproduces a high-fidelity numerical simulation.
•A physics informed neural network is designed for solving the heat diffusion equation.•An ensemble method increases accuracy of predictions and quantifies uncertainty.•A weighting heuristic automatically normalizes individual components of loss function.•Equitable convergence amongst competing minimization objectives is enforced.•Network design parameters are optimized for both accuracy and reliability.
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We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities. The large time behavior of temperature, the ...solution of the problem, is studied when initially temperature is assigned to be 0 on one medium and 1 on the other. We show that under a certain geometric condition of the configuration of the media, temperature is stabilized to a constant as time tends to infinity. We also show by examples that temperature in general oscillates and is not stabilized.
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This paper addresses the entire functions via theory of the tempered xi function. The zeros of the entire functions by the Fourier sine and cosine integrals are stud?ied in detail. The method for the ...heat-diffusion equation is proposed to structure the connection between Fourier analysis and heat-diffusion equation show the open problems in analytic number theory.
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We consider the Cauchy problem for the heat diffusion equation in the whole space consisting of three layers with different constant conductivities, where initially the upper and middle layers have ...temperature 0 and the lower layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the lower layer and the middle layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the lower layer, then the two interfaces must be parallel hyperplanes. Similar propositions hold true, either if a stationary isothermic surface is replaced by a surface with the constant flow property or if the Cauchy problem is replaced by an appropriate initial-boundary value problem.
Nous considérons le problème de Cauchy pour l'équation de diffusion de la chaleur dans tout l'espace composé de trois couches avec différentes conductivités constantes, où initialement les couches supérieure et moyenne ont la température 0 et la couche inférieure a la température 1. Dans certaines conditions appropriées, il est montré que, si l'interface entre la couche inférieure et la couche intermédiaire est une surface isotherme stationnaire ou s'il existe une surface isothermique stationnaire dans la couche intermédiaire près de la couche inférieure, alors les deux interfaces doivent être des hyperplans parallèles. Des propositions similaires sont vraies, soit si une surface isotherme stationnaire est remplacée par une surface avec la propriété d'écoulement constant ou si le problème de Cauchy est remplacé par un problème de valeur de limite initiale approprié.
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•Three small solar ponds with three different salts have been studied.•Crank Nicholson scheme has been used to solve the bi-dimensional heat diffusion equation.•Good agreement between numerical and ...experimental results have been obtained.•The salts used were NaCl, Na2CO3 and CaCl2.•CaCl2 responded more quickly than the two formers.
We consider three experimental small solar ponds (1m×1m×1m) each pond contains one of the following salts, namely NaCl, Na2CO3 and CaCl2 respectively. The thermal behavior of these three ponds is being investigated numerically and experimentally over a period of 28days. A bi-dimensional heat diffusion equation has been resolved numerically using the finite differences scheme of Crank–Nicholson. The experimental results show a good agreement with those obtained by simulation with an error of less than 1.5%. This study shows that CaCl2 pond responds thermally more quickly than the two other ponds without reaching saturation. This extends further the applications scope with possible higher temperature despite relatively its higher cost which deserves further investigation.
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The importance of energy-saving and correct design is obvious for energy efficiency. Correct design means that before construction considerable things, such as orientation or isolation decisions, ...need to be made. This study gives a mathematical model of the nonstationary energy consumption calculation problems. The model is well-posedness in Holder spaces of the mixed one-dimensional parabolic problem with Robin boundary conditions. In this study, an effective numerical method is also developed for energy consumption calculation which is related to this mathematical model. The three case problems are taken to test this numerical method. The dynamic model results have been compared with the previous finite-difference or steady-state solutions. The study also aims to develop a mathematical model in which the result can be found at any time.
In this article the theory of the supertrigonometric and superhyperbolic
functions associated with the J and H functions are proposed for the first
time. The series representation for the ...heat-diffusion equations are also
given by using the J and H functions. The results are efficient and accurate
for the description for the solutions of the PDE in mathematical physics.
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The purpose of this paper is to prove a priori error estimates for the completely discretized problem of the dual mixed method for the non-stationary heat diffusion equation in a polygonal domain of ...R2. Due to the geometric singularities of the domain, the exact solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces in our analysis. To obtain optimal convergence rates of the discrete solutions, it is necessary to refine adequately the considered meshings near the reentrant corners of the polygonal domain. In a previous work, using the Raviart–Thomas vectorfields of degree 0 for the discretization of the heat flux density vector, we have obtained a priori error estimates of order 1 for the semi-discrete solutions of this problem. In our actual paper, we complete the discretization of the problem in time by using Euler’s implicit scheme, and we obtain optimal error estimates of order 1, in time and space. Numerical results are given to illustrate this improvement of the convergence orders.
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