Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

In this paper, we extend the invariant subspace method to coupled fractional higher order nonlinear partial differential equations. In particular, we consider a two coupled fractional partial differential equations of the form [Formula: see text] where [Formula: see text], [Formula: see text] are third-order nonlinear differential operators with quadratic nonlinearities and study the four-dimensional subspaces admitted by them. Using the obtained invariant subspaces we find explicit solutions to many two coupled fractional partial differential equations including the time-fractional equivalent of potential Korteweg–de Vries equation, reduced Super Korteweg–de Vries equation, supersymmetric extension of Korteweg–de Vries equation and generalized Drinfeld–Sokolov equation.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

A new treatment for homotopy perturbation method is introduced. The new treatment is called He‐Laplace method which is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The method is implemented on linear and nonlinear partial differential equations. It is found that the proposed scheme provides the solution without any discretization or restrictive assumptions and avoids the round‐off errors.

Recently a new method for solving linear and integrable nonlinear partial differential equations (PDEs) in two dimensions has been introduced. For linear PDEs, this method involves the construction of appropriate integral representations for both the solution and the associated spectral function. Here we present an alternative approach for constructing these integral representations. This approach is based on the introduction of what we call a fundamental differential form, and on a reformulation of Green's formula. The new approach is illustrated for the Laplace and the modified Helmholtz equations in a convex polygon, for a dispersive evolution equation with spatial derivatives of arbitrary order on the half-line, and for the heat equation in a domain involving a moving boundary. In addition we show that the new approach can be used to construct appropriate integral representations for multidimensional linear PDEs.

Solution of nonlinear partial differential equations from base equations

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

This study proposes a new semi-analytical design and implementation method for nonlinear partial differential equation (PDE) control of a flexible manipulator. The proposed scheme considers the effects of the boundary input force and gravity on the payload, which results in non-homogenous boundary conditions. This objective is achieved based on a model transformation scheme for homogenizing boundary conditions, obtaining semi-analytical solutions for the corresponding PDE model. Model transformation is assigned as a hybrid exponential–polynomial function whose coefficients are conveniently calculable without the need for any additional boundary condition measurements. This eliminates the need to use intensive numerical solvers—for example, methods based on finite element analysis—and allows the implementation of sophisticated PDE control schemes considering fully nonlinear PDE models with high computation speed. The presented controller is robust to parametric model uncertainty due to its adaptive design. The precision and efficiency of calculating distributed states using the proposed model transformation are demonstrated based on experimental data for the flexible manipulator with respect to the ground truth camera-based motion capture system. Model transformation is also numerically implemented for the proposed nonlinear endpoint control method based on the original PDE model. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper investigates the difficulty of obtaining data describing the flexible manipulator pose required for precise control and analysis and proposes a computationally efficient method to overcome this issue.

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

An adaptive wavelet method for nonlinear partial differential equations with applications to dynamic damage modeling

The companion volume of this book, ‘Analytic methods for partial differential equations’, is concerned with solution of partial differential equations using classical methods which result in analytic solutions. These equations result when almost any physical situation is modelled, ranging from fluid mechanics problems, through electromagnetic problems to models of the economy. Some specific and fundamental problems were highlighted in the earlier volume, namely the three major classes of linear second order partial differential equations. The heat equation, the wave equation and Laplace’s equation will form a basis for study from a numerical point of view for the same reason as they did in the analytic case. That is, the three equations are the canonical forms to which any quasi-linear second order equation may be reduced using the characteristic transformation.KeywordsPartial Differential EquationCanonical FormSpectral RadiusLinear Algebraic EquationBackground MathematicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.