Class Of Partial Differential Equations Research Articles

A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

Abstract We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman–Kac formula, where the underlying stochastic processes satisfy variational inequalities modeling elasto-plastic and obstacle oscillators. We then focus on solving them numerically The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.

On solvability of a quasi-elliptic partial differential equations

In this paper we study the existence and smoothness of solution of one class of partial differential equation of higher order in the bounded domain $${G \subset {R^{n}}}$$ satisfying the flexile $$\lambda $$ - horn condition are studied.

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.

Numerical solutions of fuzzy time fractional advection‐diffusion equations in double parametric form of fuzzy number

Fractional partial differential equations are a generalization of classical partial differential equations which can, in certain circumstances, give a better description of certain phenomena. In this paper, two implicit finite difference schemes are developed, analyzed, and applied to solve an initial boundary value problem involving fuzzy time fractional advection‐diffusion equation with fractional order 0<α≤1. The fuzziness of the problem considered appears in the initial and boundary conditions. A computational mechanism is presented based on double parametric form of fuzzy number to transfer the problem from uncertain to crisp form. The stability of the proposed schemes is analyzed by means of the Von Neumann method and were found to be unconditionally stable. The scheme was applied to an example to illustrate the feasibility.

Inverse Problems for a Class of Degenerate Evolution Equations with Riemann - Liouville Derivative

Abstract Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

STRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEs

The aim of the present work is the introduction of a viscosity type solution, called $strong$-$viscosity$ $solution$ emphasizing also a similarity with the existing notion of $strong$ $solution$ in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.

Propagation principle for parabolic H-measures

We extend results obtained by Francfort (An introduction to H-measures and their applications. Variational problems in materials science. Birkhauser, Basel, pp 85–110, 2006) to parabolic H-measures developed by Antonic and Lazar (J Funct Anal 265:1190–1239, 2013). The well known theory of pseudodifferential operators is extended to parabolic classes of symbols and operators and used to obtain results applicable to a wide class of partial differential equations. The main result is the propagation principle which is then applied to the Schrodinger equation and the vibrating plate equation.

Extension of triple Laplace transform for solving fractional differential equations

In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two dimensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives.

Fuzzy-Stochastic Partial Differential Equations

We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy...

ГРУППОВЫЕ СВОЙСТВА ОДНОМЕРНОГО НЕЛИНЕЙНОГО УРАВНЕНИЯ ПОРОУПРУГОСТИ

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Feedback

Gradient extremum seeking for compensating wave actuator dynamics in cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Equations (PDEs) for extremum seeking has not been studied yet. A dynamic feedback control law based on distributed parameters is proposed by employing backstepping transformation with an appropriate target system and an adequate formulation using Neumann interconnections. Local stability and convergence to a small neighborhood of the desired (but unknown) extremum is proved by means of a Lyapunov functional and the theory of averaging in infinite dimensions. Numerical simulations illustrate the theoretical results.

Numerical Solution of a Class of Nonlinear Partial Differential Equations by Using Barycentric Interpolation Collocation Method

Partial differential equations (PDEs) are widely used in mechanics, control processes, ecological and economic systems, chemical cycling systems, and epidemiology. Although there are some numerical methods for solving PDEs, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, we give the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs. Four numerical experiments are carried out and compared with other methods; the accuracy of the numerical solution obtained by the present method is obviously improved.

Partial differential equations with nonautonomous past in Favard spaces

In this paper, we study the well-posedness of the Cauchy problem for a class of partial differential equations with nonautonomous past $$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)=Bu(t)+\Phi \left( \tilde{u}_t\right) , \quad t\ge 0,&{}\\ u(0)=u^0\in E,\quad u_0=g\in L^{p}(I,E), &{} \end{array}\right. } \end{aligned}$$ (0.1) where \(I=[-r,0]\) (finite delay) or \(I=(-\infty ,0]\) (infinite delay), \(1\le p<\infty \), B is the generator of a \(C_0\)-semigroup on a Banach space E, and \(\Phi \) is a bounded linear operator from \(W^{1,p}(I,E)\) to some Banach space \(Z_B\), an intermediate space related to the operator B, for instance the Favard space \(F_B\) of B. This work generalizes (Maniar and Voigt, in: Goldstein, Nagel, Romanelli (eds) Lecture notes in pure and applied mathematics, vol 234, Marcel Dekker, New York, pp 319–330, 2003) and continues the works (Brendle and Nagel in Discrete Contin Dyn Syst 8:1–24, 2002; Fragnelli in Bull Belg Math Soc 11:133–148, 2004; Fragnelli and Nickel in Differ Integral Equ 16:327–348, 2003; Fragnelli and Tonetto in J Math Anal Appl 289:90–99, 2004; Fragnelli, Delay equations with non autonomous past, Ph.D. thesis, Tubingen, 2002; Nickel and Rhandi in Math Nachr 278:1–13, 2005; Nguyen in J Math Anal Appl 289:301–316, 2004).

Finite-time attractivity for semilinear tempered fractional wave equations

Abstract We prove the existence and finite-time attractivity of solutions to semilinear tempered fractional wave equations with sectorial operator and superlinear nonlinearity. Our analysis is based on the α-resolvent theory, the fixed point theory for condensing maps and the local estimates of solutions. An application to a class of partial differential equations will be given.

Symmetry and separability of the neutron diffusion equation

Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.

Localized method of approximate particular solutions with polynomial basis functions

The method of particular solutions (MPS) using polynomials as the basis functions has been successfully developed for solving a large class of partial differential equations. However, when a large number of collocation points are required, the above mentioned approach is not feasible since the resultant matrix is dense and ill-conditioned. This restriction is common for global methods. One of the alternative approaches is to employ the localized scheme in which only a small number of neighboring points are being used in the solution process. As a result, the resultant matrix of the localized method is sparse and thus can be solved efficiently.In this paper, the localized method has been employed to extend the recently developed MPS using polynomial basis functions for solving large-scale science and engineering problems. Overall, the proposed approach is stable and highly accurate. Four numerical examples are presented to validate the proposed numerical method.

Numerical Solution of Two-Dimensional Variable-Order Fractional Optimal Control Problem by Generalized Polynomial Basis

This paper deals with an efficient numerical method for solving two-dimensional variable-order fractional optimal control problem. The dynamic constraint of two-dimensional variable-order fractional optimal control problem is given by the classical partial differential equations such as convection–diffusion, diffusion-wave and Burgers’ equations. The presented numerical approach is essentially based on a new class of basis functions with control parameters, called generalized polynomials, and the Lagrange multipliers method. First, generalized polynomials are introduced and an explicit formulation for their variable-order fractional operational matrix is obtained. Then, the state and control functions are expanded in terms of generalized polynomials with unknown coefficients and control parameters. By using the residual function and its 2-norm, the under consideration problem is transformed into an optimization one. Finally, the necessary conditions of optimality results in a system of algebraic equations with unknown coefficients and control parameters can be simply solved. Some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.

Reduction of the non-uniform illumination using nonlocal variational models for document image analysis

In this paper, we investigate two new reflectance and illumination decomposition models based on a nonlocal partial differential equation (PDE) applied to text images. Taking into consideration the higher regularity level of the illumination compared to the reflectance, we propose a nonlocal PDE which deals with repetitive structures and textures that characterize the text image much better compared to the classical local PDEs. The aim of this approach is to use the repetitive features of the reflectance to efficiently extract it from the non-uniform illumination. This idea is motivated by extending the range of application of the nonlocal operators to such a problem. Numerical experiments on both grayscale and color text images show the performance and strength of the proposed nonlocal PDE.

Surface permeability of porous media particles and capillary transport

.We have established previously, in a lead-in study, that the spreading of liquids in particulate porous media at low saturation levels, characteristically less than 10% of the void space, has very distinctive features in comparison to that at higher saturation levels. In particular, we have found that the dispersion process can be accurately described by a special class of partial differential equations, the super-fast non-linear diffusion equation. The results of mathematical modelling have demonstrated very good agreement with experimental observations. However, any enhancement of the accuracy and predictive power of the model, keeping in mind practical applications, requires the knowledge of the effective surface permeability of the constituent particles, which defines the global, macroscopic permeability of the particulate media. In the paper, we demonstrate how this quantity can be determined through the solution of the Laplace-Beltrami Dirichlet problem, we study this using the well-developed surface finite-element method.Graphical abstract

Robust data-driven discovery of governing physical laws with error bars.

Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier-Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.