Reduced differential transform method for solving (1 + n) – Dimensional Burgers' equation

AbstractThis paper discusses a recently developed semi-analytic technique so called the reduced differential transform method (RDTM) for solving the (1 + n) – dimensional Burgers' equation. The method considers the use of the appropriate initial or boundary conditions and finds the solution without any discretization, transformation, or restrictive assumptions. Four numerical examples are provided in order to validate the efficiency and reliability of the method and furthermore to compare its computational effectiveness with other analytical methods available in the literature

A new modified homotopy perturbation method for fractional partial differential equations with proportional delay

In this paper, we suggest and analyze a technique by combining the Shehu transform method and the homotopy perturbation method. This method is called the Shehu transform homotopy method (STHM). This method is used to solve the time-fractional partial differential equations (TFPDEs) with proportional delay. The fractional derivative is described in Caputo's sense. The solutions proposed in the series converge rapidly to the exact solution. Some examples are solved to show the STHM is easy to apply

FRDTM for numerical simulation of multi-dimensional, time-fractional model of Navier–Stokes equation

AbstractIn this paper, a new approximate solution of time-fractional order multi-dimensional Navier–Stokes equation is obtained by adopting a semi-analytical scheme: “Fractional Reduced Differential Transformation Method (FRDTM)”. Three test problems are carried out in order to validate and illustrate the efficiency of the method. The scheme is found to be very reliable, effective and efficient powerful technique to solve wide range of problems arising in engineering and sciences. The small size of computation contrary to the other schemes, is its strength

Numerical simulation of flooding from multiple sources using adaptive anisotropic unstructured meshes and machine learning methods

Over the past few decades, urban floods have been gaining more attention due to their increase in frequency. To provide reliable flooding predictions in urban areas, various numerical models have been developed to perform high-resolution flood simulations. However, the use of high-resolution meshes across the whole computational domain causes a high computational burden. In this thesis, a 2D control-volume and finite-element (DCV-FEM) flood model using adaptive unstructured mesh technology has been developed. This adaptive unstructured mesh technique enables meshes to be adapted optimally in time and space in response to the evolving flow features, thus providing sufficient mesh resolution where and when it is required. It has the advantage of capturing the details of local flows and wetting and drying front while reducing the computational cost. Complex topographic features are represented accurately during the flooding process. This adaptive unstructured mesh technique can dynamically modify (both, coarsening and refining the mesh) and adapt the mesh to achieve a desired precision, thus better capturing transient and complex flow dynamics as the flow evolves. A flooding event that happened in 2002 in Glasgow, Scotland, United Kingdom has been simulated to demonstrate the capability of the adaptive unstructured mesh flooding model. The simulations have been performed using both fixed and adaptive unstructured meshes, and then results have been compared with those published 2D and 3D results. The presented method shows that the 2D adaptive mesh model provides accurate results while having a low computational cost. The above adaptive mesh flooding model (named as Floodity) has been further developed by introducing (1) an anisotropic dynamic mesh optimization technique (anisotropic-DMO); (2) multiple flooding sources (extreme rainfall and sea-level events); and (3) a unique combination of anisotropic-DMO and high-resolution Digital Terrain Model (DTM) data. It has been applied to a densely urbanized area within Greve, Denmark. Results from MIKE 21 FM are utilized to validate our model. To assess uncertainties in model predictions, sensitivity of flooding results to extreme sea levels, rainfall and mesh resolution has been undertaken. The use of anisotropic-DMO enables us to capture high resolution topographic features (buildings, rivers and streets) only where and when is needed, thus providing improved accurate flooding prediction while reducing the computational cost. It also allows us to better capture the evolving flow features (wetting-drying fronts). To provide real-time spatio-temporal flood predictions, an integrated long short-term memory (LSTM) and reduced order model (ROM) framework has been developed. This integrated LSTM-ROM has the capability of representing the spatio-temporal distribution of floods since it takes advantage of both ROM and LSTM. To reduce the dimensional size of large spatial datasets in LSTM, the proper orthogonal decomposition (POD) and singular value decomposition (SVD) approaches are introduced. The performance of the LSTM-ROM developed here has been evaluated using Okushiri tsunami as test cases. The results obtained from the LSTM-ROM have been compared with those from the full model (Fluidity). Promising results indicate that the use of LSTM-ROM can provide the flood prediction in seconds, enabling us to provide real-time flood prediction and inform the public in a timely manner, reducing injuries and fatalities. Additionally, data-driven optimal sensing for reconstruction (DOSR) and data assimilation (DA) have been further introduced to LSTM-ROM. This linkage between modelling and experimental data/observations allows us to minimize model errors and determine uncertainties, thus improving the accuracy of modelling. It should be noting that after we introduced the DA approach, the prediction errors are significantly reduced at time levels when an assimilation procedure is conducted, which illustrates the ability of DOSR-LSTM-DA to significantly improve the model performance. By using DOSR-LSTM-DA, the predictive horizon can be extended by 3 times of the initial horizon. More importantly, the online CPU cost of using DOSR-LSTM-DA is only 1/3 of the cost required by running the full model.Open Acces

The Telegraph Equation and Its Solution by Reduced Differential Transform Method

One-dimensional second-order hyperbolic telegraph equation was formulated using Ohm’s law and solved by a recent and reliable semianalytic method, namely, the reduced differential transform method (RDTM). Using this method, it is possible to find the exact solution or a closed approximate solution of a differential equation. Three numerical examples have been carried out in order to check the effectiveness, the accuracy, and convergence of the method. The RDTM is a powerful mathematical technique for solving wide range of problems arising in science and engineering fields

Diamagnetic Stabilization of Double-tearing Modes in MHD Simulations

Double-tearing modes have been proposed as a driver of ‘off-axis sawtooth’ crashes in reverse magnetic shear tokamak configurations. The DTM consists of two nearby rational surfaces of equal safety factor that couple to produce a reconnecting mode weakly dependent on resistivity and capable of nonlinearly disrupting the annular current. In this dissertation we examine the linear and nonlinear growth of the DTM using the extended magnetohydrodynamic simulation code MRC-3d. We consider the efficacy of equilibrium diamagnetic drifts, which emerge in the presence of a pressure gradient when ion inertial physics is included, as a means of stabilizing DTM activity. In linear slab simulations we find that a differential diamagnetic drift at the two resonant surfaces is able to both interfere with the inter-surface coupling and suppress the reconnection process internal to the tearing layers. Applying these results to a m=2, n=1 DTM in cylindrical geometry, we find that asymmetries between the resonant layers and the presence of an ideal MHD mode result in stabilization being highly dependent on the location of the pressure gradient. We achieve a significant reduction in the linear DTM growth rate by locating a strong diamagnetic drift at the outer resonant surface. In nonlinear simulations we show that growth of the magnetic islands may enhance the pressure gradient near the DTM current sheets and significantly delay disruption. Only by locating a strong drift near the outer, dominant resonant surface are we able to saturate the mode and preserve the annular current ring, suggesting that the appearance of DTM activity in advanced tokamaks may depend on the details of the plasma pressure profile

Wave Propagation

A wave is one of the basic physics phenomena observed by mankind since ancient time. The wave is also one of the most-studied physics phenomena that can be well described by mathematics. The study may be the best illustration of what is “science”, which approximates the laws of nature by using human defined symbols, operators, and languages. Having a good understanding of waves and wave propagation can help us to improve the quality of life and provide a pathway for future explorations of the nature and universe. This book introduces some exciting applications and theories to those who have general interests in waves and wave propagations, and provides insights and references to those who are specialized in the areas presented in the book