Second-order Partial Differential Equations Research Articles

Travelling wave solutions in dilatant non-Newtonian thin films with second-order viscosity

Given any positive speed, we prove the existence of a travelling wave solution for a viscous incompressible thin dilatant liquid film coated with an insoluble surfactant. Relying on first principles of fluid mechanics, lubrication theory leads to a coupled system of second-order partial differential equations for the film's height and the surfactant's concentration where a second-order approximation of the viscosity function is used. The travelling wave solution is -regular and generates a heteroclinic orbit which connects two distinct equilibrium points of the associated plane dynamical system.

Uncertain Wave Equation for Vibrating String

Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.

Comparing stochastic Lotka–Volterra predator-prey models

The phenomenology of population extinction is one of the central themes in population biology which it is an inherently stochastic event. In the present investigation, we study this problem for three different stochastic models built from a single Lotka–Volterra deterministic model. More concretely, we study their mean-extinction time which satisfies the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients; hence, we can only compute numerical approximations. We suggest a finite element method using FreeFem++. Our analysis and numerical results allow us to conclude that there are important differences between the three models. These differences enable us to choose the most “natural way” to turn a the deterministic model into a stochastic model.

Compact half step approximation in exponential form for the system of 2D second-order quasi-linear elliptic partial differential equations

ABSTRACTThis paper deals with devising a novel exponential scheme, implicit in nature, using half step discretization of order four for computing the numerical solution of quasi-linear elliptic partial differential equations of the type , . Nine mesh points in a single compact cell are being used in this method. An elaborate convergence analysis of the proposed method has been worked out. The technique for a scalar equation is eventually extended to solve the systems of quasi-linear elliptic equations. The technique is then applied to noteworthy problems and computational outcomes corroborate the theoretical rate of convergence. Linear and non-linear singular problems are tackled separately ensuring the usage of nine spatial grid points of a single computing cell.

A parametric study on the dynamics of two-span roll-to-roll microcontact printing system

The recent trend towards micro-patterning of thin film webs brought stricter manufacturing tolerances than traditionally admissible vibration levels. Therefore, one has to consider roll-to-roll (R2R) systems with their system dynamics model encompassing all systems’ components, namely, the axially moving web(s), the rotating rolls and other accessories. In this study, the dynamic characteristics of an axially moving web in a two-span R2R system are investigated. Influence of geometrical parameters on the R2R dynamics and the parametric resonances of the moving web is presented. The time variations of web axial speed and web-transmitted tension are rigorously obtained by solving the web tension–roller angular speed equation, which is obtained from the conservation of mass law. The effects of system parameters such as length of the web, inertia of idler roll and radius of the rewinding/unwinding roll on the angular speed and web-transmitted tension have been discussed. The axially moving web was modelled as a string, which was mathematically represented by a second-order hyperbolic partial differential equation. The effect of the frequency of the disturbance at the roll on the web dynamics is also discussed. The transverse vibration response at selected points on the web shows higher frequency fluctuations corresponding to lower transport axial speed. It is noted that instability in the transverse response will occur when the first and second frequencies of the oscillation converge.

Stabilization of high order cut finite element methods on surfaces

Abstract We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in ${\mathbb{R}}^d$, with codimension $d-n$. We also state the properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.

Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions. As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.

One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes

A comprehensive analysis of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates is performed. One of the representations of these equations in Lagrangian coordinates is given by a single second-order partial differential equation. Symmetries of this equation are analyzed using the entropy for the group classification. Noether’s theorem is applied for constructing conservation laws. The obtained conservation laws are represented in the gas dynamics variables in Lagrangian coordinates and in Eulerian coordinates as well. Invariant and conservative difference schemes are discussed for the basic adiabatic case.

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first- and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23–37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first- and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

Nonlocal Boundary-Value Problem for a Second-Order Partial Differential Equation in an Unbounded Strip

The conditions of well-posedness of nonlocal boundary-value problems are established for a second-order linear partial differential equation in an unbounded strip in the case where the real parts of the roots of its characteristic equation are different and nonzero.

Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory

We are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation, which describes the dynamics of many viscoelastic systems and covers the well-known Timoshenko system. We present an optimal energy decay estimate under very weak conditions. More specifically, we prove that the energy of the system decays to zero at least at the rate of t−1 only under the condition that the related kernel functions are non-increasing and integrable. Moreover, we apply our abstract results to three classes of coupled systems of second-order partial differential equations arising in viscoelasticity.

A quadratic spline collocation method for the Dirichlet biharmonic problem

A new method based on quadratic spline collocation is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves the solution of an auxiliary biharmonic problem using fast Fourier transforms and the solution of a nonsymmetric Schur complement system using preconditioned BICGSTAB, at a total cost of $N^{2} \log N$ on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimality of the global accuracy of the method and also superconvergence results, in particular, third-order accuracy in the $L^{\infty }$ norm of the solution and its fourth-order accuracy at the partition nodes and the collocation points.

Leading and Second Order Homogenization of an Elastic Scattering Problem for Highly Oscillating Anisotropic Medium

We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We further pursue a higher-order homogenization with the help of complimentary boundary correctors and provide a detailed analysis on the rate of higher-order convergence. Finally we provide preliminary numerical examples to demonstrate the higher-order homogenization.

Method of Integral Equations for Studying the Solvability of Boundary Value Problems for the System of Nonlinear Differential Equations of the Theory of Timoshenko Type Shallow Inhomogeneous Shells

The solvability of the boundary value problem for a system of second-order nonlinear partial differential equations with given boundary conditions which describes the equilibrium of elastic inhomogeneous shallow shells with free edges in the framework of the Timoshenko shear model is considered. The boundary value problem is reduced to a single nonlinear equation whose solvability is established by using the contraction mapping principle.

Crossed-laser-beam solutions for the Klein-Gordon equation

We present analytic solutions of the Klein-Gordon equation for an electron of momentum $p$ in $N$ crossed laser beams ${\mathbf{A}}_{j}\left({\ensuremath{\varphi}}_{j}\right)$ with phases ${\ensuremath{\varphi}}_{j}={\ensuremath{\omega}}_{j}t\ensuremath{-}{\mathbf{k}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r}+{\ensuremath{\delta}}_{j}.$ The solutions are of the form ${\mathrm{\ensuremath{\Psi}}}_{p}={e}^{ipx}\phantom{\rule{0.16em}{0ex}}{\mathit{\ensuremath{\digamma}}}_{p}\left(\mathbf{r},t\right).$ The determination of the distortion factor ${\mathit{\ensuremath{\digamma}}}_{p}$ is pursued within a method of ``auxiliary variables,'' which uses the ${\ensuremath{\varphi}}_{j}$ as variables, ${\mathit{\ensuremath{\digamma}}}_{p}\left(\mathbf{r},t\right)\ensuremath{\equiv}{\mathit{\ensuremath{\digamma}}}_{p}\left({\ensuremath{\varphi}}_{1},...,{\ensuremath{\varphi}}_{N}\right)$. The equation for ${\mathit{\ensuremath{\digamma}}}_{p}\left({\ensuremath{\varphi}}_{1},...,{\ensuremath{\varphi}}_{N}\right)$ is a linear second-order partial differential equation and it does not appear to be soluble analytically exactly. However, the second-order derivative terms are multiplied by coeffients that are small with respect to those of the first-order derivatives, if (A) the angles between all beams are small or (B) for all beams we have ${\ensuremath{\omega}}_{j}/m{c}^{2}\ensuremath{\ll}1.$ In the second case, their neglect would be quite justified for frequencies up into the x-ray range. With this approximation the equation would reduce to a first-order derivatives ``reduced equation.'' Mathematically, however, this cannot be done without further analysis. This is because we are in a situation typical of singular perturbations theory, in which the exact, perturbed solution might not be connected continuously to the unperturbed one. Nevertheless, on the basis of exactly soluble models (see the Appendix), we argue that the approximation is justified in our case and proceed to solve it. It turns out that the reduced equation can be solved exactly for certain crossed-beam geometries of interest. We consider first the case of laser pulses of arbitrary shape. The most general geometry we solve is that in which all beams have coplanar propagation directions and the fields are linearly polarized perpendicular to the propagation plane, except possibly for two that can be oblique and elliptically polarized. For two beams $(N=2)$ this covers the most general case possible. As an application, we calculate the closed-form solution for Gaussian-pulse crossed beams. Next, we treat the case of monochromatic beams as a limit of the laser-pulse case and derive closed-form solutions for some geometries, including standing waves. We then discuss the effect of the passage of crossed beams over an electronic wave packet and show that its momentum distribution is not modified and no pair production is possible (in the reduced equation approximation). We also show that the final wave packet displays the classical ponderomotive Lorentz shift, if low relativistic momenta are involved. The Appendix deals with soluble models of the exact equation for ${\mathit{\ensuremath{\digamma}}}_{p}$ which have the salient features of the original. For these models, the reduced equation solution is indeed a valid approximation to the exact one, if conditions similar to (A) or (B) above are met.

An adaptive second-order partial differential equation based on TV equation and p-Laplacian equation for image denoising

This paper introduces an adaptive diffusion partial differential equation (PDE) for noise removal, which combines a total variation (TV) term and a p-Laplacian (1 < p ≤ 2) term. Utilizing the edge indicator, we can adaptively control the diffusion model, which alternates between the TV and the p-Laplacian(1 < p ≤ 2) in accordance with the image feature. The main advantage of the proposed model is able to alleviate the staircase effect in smooth regions and preserve edges while removing the noise. The existence of a weak solution of the proposed model is proved. Experimental results confirm the performance of the proposed method with regard to peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) and visual quality.

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.

Strong approximation of monotone stochastic partial differential equations driven by white noise

Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

Analysis of the vanishing moment method and its finite element approximations for second-order linear elliptic PDEs in non-divergence form

This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.