Showing papers on "Partial differential equation published in 2015"

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Abstract: This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Reduced Basis Methods for Partial Differential Equations: An Introduction

Abstract: 1 Introduction.- 2 Representative problems: analysis and (high-fidelity) approximation.- 3 Getting parameters into play.- 4 RB method: basic principle, basic properties.- 5 Construction of reduced basis spaces.- 6 Algebraic and geometrical structure.- 7 RB method in actions.- 8 Extension to nonaffine problems.- 9 Extension to nonlinear problems.- 10 Reduction and control: a natural interplay.- 11 Further extensions.- 12 Appendix A Elements of functional analysis.

CutFEM: Discretizing geometry and partial differential equations

Abstract: We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

Paracontrolled distributions and singular PDEs

Abstract: We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

Numerical Methods for Fractional Calculus

Abstract: Introduction to Fractional Calculus Fractional Integrals and Derivatives Some Other Properties of Fractional Derivatives Some Other Fractional Derivatives and Extensions Physical Meanings Fractional Initial and Boundary Problems Numerical Methods for Fractional Integral and Derivatives Approximations to Fractional Integrals Approximations to Riemann-Liouville Derivatives Approximations to Caputo Derivatives Approximation to Riesz Derivatives Matrix Approach Short Memory Principle Other Approaches Numerical Methods for Fractional Ordinary Differential Equations Introduction Direct Methods Integration Methods Fractional Linear Multistep Methods Finite Difference Methods for Fractional Partial Differential Equations Introduction One-Dimensional Time-Fractional Equations One-Dimensional Space-Fractional Differential Equations One-Dimensional Time-Space Fractional Differential Equations Fractional Differential Equations in Two Space Dimensions Galerkin Finite Element Methods for Fractional Partial Differential Equations Mathematical Preliminaries Galerkin FEM for Stationary Fractional Advection Dispersion Equation Galerkin FEM for Space-Fractional Diffusion Equation Galerkin FEM for Time-Fractional Differential Equations Galerkin FEM for Time-Space Fractional Differential Equations Bibliography Index

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis

Abstract: Isotropic and Anisotropic Function Spaces Lebesgue and Sobolev Spaces with Variable Exponent History of function spaces with variable exponent Lebesgue spaces with variable exponent Sobolev spaces with variable exponent Dirichlet energies and Euler-Lagrange equations Lavrentiev phenomenon Anisotropic function spaces Orlicz spaces Variational Analysis of Problems with Variable Exponents Nonlinear Degenerate Problems in Non-Newtonian Fluids Physical motivation A boundary value problem with nonhomogeneous differential operator Nonlinear eigenvalue problems with two variable exponents A sublinear perturbation of the eigenvalue problem associated to the Laplace operator Variable exponents versus Morse theory and local linking The Caffarelli-Kohn-Nirenberg inequality with variable exponent Spectral Theory for Differential Operators with Variable Exponent Continuous spectrum for differential operators with two variable exponents A nonlinear eigenvalue problem with three variable exponents and lack of compactness Concentration phenomena: the case of several variable exponents and indefinite potential Anisotropic problems with lack of compactness and nonlinear boundary condition Nonlinear Problems in Orlicz-Sobolev Spaces Existence and multiplicity of solutions A continuous spectrum for nonhomogeneous operators Nonlinear eigenvalue problems with indefinite potential Multiple solutions in Orlicz-Sobolev spaces Neumann problems in Orlicz-Sobolev spaces Anisotropic Problems: Continuous and Discrete Anisotropic Problems Eigenvalue problems for anisotropic elliptic equations Combined effects in anisotropic elliptic equations Anisotropic problems with no-flux boundary condition Bifurcation for a singular problem modelling the equilibrium of anisotropic continuous media Difference Equations with Variable Exponent Eigenvalue problems associated to anisotropic difference operators Homoclinic solutions of difference equations with variable exponents Low-energy solutions for discrete anisotropic equations Appendix A: Ekeland Variational Principle Appendix B: Mountain Pass Theorem Bibliography Index A Glossary is included at the end of each chapter.

Local Fractional Integral Transforms and Their Applications

Abstract: Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms.Provides applications of local fractional Fourier SeriesDiscusses definitions for local fractional Laplace transformsExplains local fractional Laplace transforms coupled with analytical methods

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Abstract: The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

Solving Boundary Integral Problems with BEM

Abstract: Many important partial differential equation problems in homogeneous media, such as those of acoustic or electromagnetic wave propagation, can be represented in the form of integral equations on the boundary of the domain of interest. In order to solve such problems, the boundary element method (BEM) can be applied. The advantage compared to domain-discretisation-based methods such as finite element methods is that only a discretisation of the boundary is necessary, which significantly reduces the number of unknowns. Yet, BEM formulations are much more difficult to implement than finite element methods. In this article, we present BEMpp, a novel open-source library for the solution of boundary integral equations for Laplace, Helmholtz and Maxwell problems in three space dimensions. BEMpp is a Cpp library with Python bindings for all important features, making it possible to integrate the library into other Cpp projects or to use it directly via Python scripts. The internal structure and design decisions for BEMpp are discussed. Several examples are presented to demonstrate the performance of the library for larger problems.

A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application

Abstract: The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Backlund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Backlund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics.

Second order mean field games with degenerate diffusion and local coupling

Abstract: We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

Mixed convection flow of MHD Eyring-Powell nanofluid over a stretching sheet: A numerical study

Abstract: In the present analysis incompressible two dimensional mixed convection flow of MHD Eyring-Powell nanofluid over a stretching sheet is investigated numerically. The governing highly nonlinear partial differential equations are converted into ordinary differential equations by using a similarity approach. Numerical solutions of the nonlinear ordinary differential equations are found by using a shooting method. Effects of various parameters are displayed graphically for velocity, temperature and concentration profiles. Also quantities of practical interest i.e skin friction coefficient, Nusselt number and Sherwood number are presented graphically and tabularly.

Methods of Nonlinear Analysis: Applications to Differential Equations

Abstract: Preface.- 1 Preliminaries.- 2 Properties of Linear and Nonlinear Operators.- 3 Abstract Integral and Differential Calculus.- 4 Local Properties of Differentiable Mappings.- 5 Topological and Monotonicity Methods.- 6 Variational Methods.- 7 Boundary Value Problems for Partial Differential Equations.- Summary of Methods.- Typical Applications.- Comparison of Bifurcation Results.- List of Symbols.- Index.- Bibliography.

Numerical Methods for Nonlinear Partial Differential Equations

Abstract: 1. Introduction.- Part I: Analytical and Numerical Foundations.- 2. Analytical Background.- 3. FEM for Linear Problems.- 4. Concepts for Discretized Problems.- Part II: Approximation of Classical Formulations.- 5. The Obstacle Problem.- 6. The Allen-Cahn Equation.- 7. Harmonic Maps.- 8. Bending Problems.- Part III: Methods for Extended Formulations.- 9. Nonconvexity and Microstructure.- 10. Free Discontinuities.- 11. Elastoplasticity.- Auxiliary Routines.- Frequently Used Notation.- Index.

The Active Disturbance Rejection Control to Stabilization for Multi-Dimensional Wave Equation With Boundary Control Matched Disturbance

Abstract: In this paper, we consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi-dimensional system by infinitely many test functions. The disturbance is canceled in the feedback loop together with a collocated stabilizing controller. All subsystems in the closed-loop are shown to be asymptotically stable. In particular, the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature. The overall picture of the ADRC in dealing with the disturbance for multi-dimensional partial differential equation is presented through this system. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Abstract: Summary We present a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M3) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract, even from these results with real data, the numerical trends indicated by the theoretical convergence and stability analyses. Copyright © 2014 John Wiley & Sons, Ltd.

Two-Sided Projection Methods for Nonlinear Model Order Reduction

Abstract: In this paper, we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching. Using basic tensor calculus, we propose a computationally efficient way of computing reduced-order models. We further extend the idea of two-sided interpolation methods to this more general setting by employing the tensor structure of the Hessian. We investigate the use of oblique projections in order to preserve important system properties such as stability. We test one-sided and two-sided projection methods for different semi-discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition (POD).

A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Abstract: Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality...

Study on blood flow containing nanoparticles through porous arteries in presence of magnetic field using analytical methods

Abstract: In this paper, flow analysis for a third grade non-Newtonian blood in porous arteries in presence of magnetic field is simulated analytically and numerically. Blood is considered as the third grade non-Newtonian fluid containing nanoparticles. Collocation Method (CM) and Optimal Homotopy Asymptotic Method (OHAM) are used to solve the Partial Differential Equation (PDE) governing equation which a good agreement between them was observed in the results. The influences of the some physical parameters such as Brownian motion parameter, pressure gradient and thermophoresis parameter, etc. on temperature, velocity and nanoparticles concentration profiles are considered. For instance, increasing the thermophoresis parameter (Nt) caused an increase in temperature values in whole domain and an increase in nanoparticles concentration near the inner wall.

Dynamic Modeling and Interaction of Hybrid Natural Gas and Electricity Supply System in Microgrid

Abstract: Natural gas (NG) network and electric network are becoming tightly integrated by microturbines in the microgrid. Interactions between these two networks are not well captured by the traditional microturbine (MT) models. To address this issue, two improved models for single-shaft MT and split-shaft MT are proposed in this paper. In addition, dynamic models of the hybrid natural gas and electricity system (HGES) are developed for the analysis of their interactions. Dynamic behaviors of natural gas in pipes are described by partial differential equations (PDEs), while the electric network is described by differential algebraic equations (DAEs). So the overall network is a typical two-time scale dynamic system. Numerical studies indicate that the two-time scale algorithm is faster and can capture the interactions between the two networks. The results also show the HGES with a single-shaft MT is a weakly coupled system in which disturbances in the two networks mainly influence the dc link voltage of the MT, while the split-shaft MT is a strongly coupled system where the impact of an event will affect both networks.