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Numerical Solution of Burgers\u27 equation arising in Longitudinal Dispersion Phenomena in Fluid Flow through Porous Media by Crank-Nicolson Scheme

The present paper discusses the numerical solution of the Burgers’ equation arising in longitudinal dispersion phenomenon in fluid flow through porous media. In the porous medium pure water, salt water or contaminated water disperse in longitudinal direction gives rise to a non-linear partial differential equation in the form of Burgers’ equation. The equation is solved by using Crank-Nicolson finite difference scheme with appropriate initial and boundary conditions. The longitudinal dispersion phenomenon may be miscible or immiscible fluid flow through porous media. The problem of miscible displacement can be seen in the coastal areas, where fresh water beds are gradually displaced by sea water. Longitudinal dispersion phenomenon plays an important role to control salinity of the soil in western seashore region of the Gujarat state in India. To control salinity, the government of Gujarat has developed many check dams from where contaminated water diverted and poured to the soil of the farms, where the crops of cumin seed (jeera), fennel (saunf) and other grains are grown. In this region due to the infiltration of this infiltered water, free surface of sweet water table rises, consequently, saline seawater cannot cross the threshold in the nearby area of the seashore. In such a way, the dispersion of contaminated water plays key role to solve salinity problem. The immiscible dispersion also plays an important role in petroleum engineering during secondary oil recovery process, in which water or gas is injected in oil formatted area to drive the oil towards production well. An unconditionally stable Crank-Nicolson finite difference scheme has been employed to find the concentration C(X, T) of salty or contaminated water dispersion in uni-direction. The outcome is consistent with physical phenomenon of longitudinal dispersion in miscible fluid flow through porous media. It is concluded, that the concentration C(X, T) decreases as distance X as well as time T increases. The tables and graphs are developed by using MATLAB coding

Physarum Powered Differentiable Linear Programming Layers and Applications

Consider a learning algorithm, which involves an internal call to an optimization routine such as a generalized eigenvalue problem, a cone programming problem or even sorting. Integrating such a method as layers within a trainable deep network in a numerically stable way is not simple -- for instance, only recently, strategies have emerged for eigendecomposition and differentiable sorting. We propose an efficient and differentiable solver for general linear programming problems which can be used in a plug and play manner within deep neural networks as a layer. Our development is inspired by a fascinating but not widely used link between dynamics of slime mold (physarum) and mathematical optimization schemes such as steepest descent. We describe our development and demonstrate the use of our solver in a video object segmentation task and meta-learning for few-shot learning. We review the relevant known results and provide a technical analysis describing its applicability for our use cases. Our solver performs comparably with a customized projected gradient descent method on the first task and outperforms the very recently proposed differentiable CVXPY solver on the second task. Experiments show that our solver converges quickly without the need for a feasible initial point. Interestingly, our scheme is easy to implement and can easily serve as layers whenever a learning procedure needs a fast approximate solution to a LP, within a larger network

Determination of TL Kinetic parameters of a phosphor

Present paper reports the methods of evaluating the kinetic parameters like trap depth, frequency factor etc. by using the Glow Curve of a phosphor. Peak shape method is found to be suitable amongst all reported methods. This method provides the nearest possible values of all studied kinetic parameters which are discussed in details. MS -Excel sheet is prepared for calculation