ADI Method for Two-dimension and Second Order Partial Differential Equation with Variable Coefficients in Piecewise Rectangle Domain

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher- order time discretization and combine it with the ADI methods. We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results. AMS subject classifications: 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25

Based on a Riccati equation and a symbolic computation system-Maple, a generalized Riccati equation expansion method is presented for constructing soliton-like solutions and periodic form solutions for some nonlinear evolution equations (NEEs) or NEEs with variable coefficients. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. We study a (2+1)-dimensional general nonintegrable KdV equation, a KdV equation with variable coefficients. As a result, rich new families of exact solutions, including the non-travelling wave's and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained. When setting the arbitrary functions in some solutions be equal to special constants or special functions, the solitary wave solutions can be recovered.

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

Collection of Solved Nonlinear Problems for Remote Shaping and Patterning of Liquid Structures on Flat and Curved Substrates by Electric and Thermal Fields

A novel (geometrical) parametrization procedure using solutions to a suitably chosen fourth order partial differential equation is used to define a class of airplane configurations. Inclusive in this definition are surface grids, volume grids, and grid sensitivity. The general airplane configuration has wing, fuselage, vertical tail and horizontal tail. The design variables are incorporated into the boundary conditions, and the solution is expressed as a Fourier series. The fuselage has circular cross section, and the radius is an algebraic function of four design parameters and an independent computational variable. Volume grids are obtained through an application of the Control Point Form method. A graphic interface software is developed which dynamically changes the surface of the airplane configuration with the change in input design variable. The software is made user friendly and is targeted towards the initial conceptual development of any aerodynamic configurations. Grid sensitivity with respect to surface design parameters and aerodynamic sensitivity coefficients based on potential flow is obtained using an Automatic Differentiation precompiler software tool ADIFOR. Aerodynamic shape optimization of the complete aircraft with twenty four design variables is performed. Unstructured and structured volume grids and Euler solutions are obtained with standard software to demonstrate the feasibility of the new surface definition.

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

The presence of noise in images is unavoidable.It may be formed by image formation process,image recording,image transmission etc.These random distortions make it difficult to perform any required image processing task.High quality noise free remote sensing images are necessary for various applications.Therefore, removal of noise is necessary. The purpose of image denoising is to preserve edges as far as possible while removing noise, making the resulting images approximate the ideal image.This paper proposes a method for denoising remote sensing images using a combination of second order and fourth order partial differential equations.The advantages of both second order and fourth order partial differential equations are utilized here.The image is denoised using second order partial differential equations , fourth order partial differential equations and the combination of both.A comparison of the three methods is also presented.The proposed algorithm smooth out more noise and conserve more detail in the denoising process.

SIR models with directed diffusions are important in describing the population movement. However, efficient numerical simulations of such systems of fully nonlinear second order partial differential equations (PDEs) are challenging. They are often mixed type PDEs with ill-posed or degenerate components. The solutions may develop singularities along with time evolution. Stiffness due to nonlinear diffusions in the system gives strict constraints in time step sizes for numerical methods. In this paper, we design efficient Krylov implicit integration factor (IIF) Weighted Essentially Non-Oscillatory (WENO) method to solve SIR models with directed diffusions. Numerical experiments are performed to show the good accuracy and stability of the method. Singularities in the solutions are resolved stably and sharply by the WENO approximations in the scheme. Unlike a usual implicit method for solving stiff nonlinear PDEs, the Krylov IIF WENO method avoids solving large coupled nonlinear algebraic systems at every time step. Large time step size computations are achieved for solving the fully nonlinear second-order PDEs, namely, the time step size is proportional to the spatial grid size as that for solving a pure hyperbolic PDE. Two biologically interesting cases are simulated by the developed scheme to study the finite-time blow-up time and location or discontinuity locations in the solution of the SIR model.

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

Over the last couple of decades the demand for high precision and enhanced performance of physical systems has been steadily increasing. This demand often results in miniaturization and complex design, thus increasing the need for complex nonlinear control methods. Some of the state of the art nonlinear methods are stymied by the requirement of full state information, model and parameter uncertainties, mathematical complexity, etc. For many scenarios it is nearly impossible to consider all the uncertainties during the design of a feedback controller. Additionally, while designing a modelbased nonlinear control there is no standard mechanism to incorporate performance measures. Some of the mentioned issues can be addressed by using online learning. Animals and humans have the ability to share, explore, act or respond, memorize the outcome and repeat the task to achieve a better outcome when they encounter the same or a similar scenario. This is called learning from interaction. One instance of this approach is reinforcement learning (RL). However, RL methods are hindered by the curse of dimensionality, non-interpretability and non-monotonic convergence of the learning algorithms. This can be attributed to the intrinsic characteristics of RL, as it is a modelfree approach and hence no standard mechanism exists to incorporate a priori model information. In this thesis, learning methods are proposed which explicitly use the available system knowledge. This can be seen as a new class of approaches that bridge model-based and model-free methods. These methods can address some of the hurdles mentioned earlier. For example, i) a prior system information can speed up the learning, ii) new control objectives can be achieved which otherwise would be extremely difficult to attain using only model-based methods, iii) physical meaning can be attributed to the learned controller. The developed approach is as follows: themodel of the given physical system is represented in the port-Hamiltonian (PH) form. For the system dynamics in PH form a passivity-based control (PBC) law is formulated, which often requires the solution to a set of partial differential equations (PDEs). Instead of finding an analytical solution, the PBC control law is parameterized using an unknown parameter vector. Then, by using a variation of the standard actor-critic learning algorithm, the unknown parameters can be learned online. Using the principles of stochastic approximation theory, a proof of convergence for the developed method is shown. The proposedmethods are evaluated for the stabilization and regulation ofmechanical and electro-mechanical systems. The simulation and experimental results show comparable learning curves. In the final part of the thesis a novel integral reinforcement learning approach is developed to solve for the optimal output tracking control problem for a set of linear heterogeneous multi-agent systems. Unlike existing methods, this approach does not need to solve either the output regulator equation or requires a p-copy of the leader’s dynamics in the agent’s control law. A detailed numerical evaluation has been conducted to show the feasibility of the developed method.

A grid based ADI method for the problem of two phase solidification

In this paper we find the complete solution of some kinds of linear third order partial differential equations of variable coefficients with three independent variables which have the general form Where A,B,…,T are variable coefficients . By use the some assumptions will transform the above equation to thenonlinear second order ordinary differential equations

A shape design system using volumetric implicit PDEs