Second-order Parabolic Equations Research Articles

Feynman Formulas for Solutions to Evolution Equations in Domains of Multidimensional Ramified Surfaces

Solutions of second-order parabolic differential equations for functions defined in domains of a K ramified surface in the class L2(K) are obtained. With the help of Chernoff’s theorem, such solutions (if they exist) can be represented in the form of Lagrangian Feynman formulas, i.e., in the form of limits of integrals over Cartesian powers of the configuration space as the number of factors tends to infinity.

Cauchy problem solutions for plane parabolic equations: stabilization velocity

ABSTRACTA logarithmic with respect to t estimate is obtained for the solution of the Cauchy problem for plane second-order parabolic equations with a zero-order term; the estimate is uniform with respect to x on each compact set.

Galerkin-Chebyshev Pseudo Spectral Method and a Split Step New Approach for a Class of Two dimensional Semi-linear Parabolic Equations of Second Order

Abstract In this paper, we use a time splitting method with higher-order accuracy for the solutions (in space variables) of a class of two-dimensional semi-linear parabolic equations. Galerkin-Chebyshev pseudo spectral method is used for discretization of the spatial derivatives, and implicit Euler method is used for temporal discretization. In addition, we use this novel method to solve the well-known semi-linear Poisson-Boltzmann (PB) model equation and obtain solutions with higher-order accuracy. Furthermore, we compare the results obtained by our method for the semi-linear parabolic equation with the available analytical results in the literature for some special cases, and found excellent agreement. Furthermore, our new technique is also applicable for three-dimensional problems.

C 1+α-Regularity of Viscosity Solutions of General Nonlinear Parabolic Equations

We establish the existence of C1+α-regular viscosity solutions of fully nonlinear secondorder parabolic equations like ∂ t u+H(v,Dv,D2v, t, x) = 0 allowing the main part of H to have small discontinuities with respect to x and be just measurable in t.

Stochastic and partial differential equations on non-smooth time-dependent domains

In this article, we consider non-smooth time-dependent domains whose boundary is W1,p in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.

Numerical solution of optimal control problem for a model of tumour growth with drug application

ABSTRACTIn this study, a combination of spectral and fixed point methods is used to solve an optimal control problem for a model of tumour growth. The growth of tumour is modelled using three first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the optimal control problem, four control variables are employed to control the concentration of nutrient and drug on the boundary and inside the tumour. Since the problem is nonlinear, applying the fixed point method, in each step of iteration, the problem is changed to a linear one and the parabolic equations are solved using the spectral method. The convergence and stability of method are proven. Some examples are considered to illustrate the efficiency of method. Finally, some figures are provided to reflect the effects of control on the densities of tumourcells.

Application of fixed point-collocation method for solving an optimal control problem of a parabolic–hyperbolic free boundary problem modeling the growth of tumor with drug application

In this paper, employing a fixed point-collocation method, we solve an optimal control problem for a model of tumor growth with drug application. This model is a free boundary problem and consists of five time-dependent partial differential equations including three different first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the mentioned optimal control problem, the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells. In this study, applying the fixed point method, we construct a sequence converging to the solution of the optimal control problem. In each step of the fixed point iteration, the problem changes to a linear one and the parabolic equations are solved using the collocation method. The stability of the method is also proved. Some examples are considered to illustrate the efficiency of method.

Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. I. Explicit Methods

We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Norsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numerical and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.

Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

An Ensemble Algorithm for Numerical Solutions to Deterministic and Random Parabolic PDEs

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could be subject to different diffusion coefficients, initial conditions, boundary conditions, and body forces. The proposed algorithm leads to a single discrete system for the group with multiple right-hand-side vectors by introducing an ensemble average of the diffusion coefficient functions and using a new semi-implicit time integration method. The system could be solved more efficiently than multiple linear systems with a single right-hand-side vector. We first apply the algorithm to deterministic parabolic PDEs and derive a rigorous error estimate that shows the scheme is first-order accurate in time and is optimally accurate in space. We then extend it to find stochastic solutions of parabolic PDEs with random coefficients and put forth an ensemble-based Monte Carlo met...

Partial Schauder estimates for ellipticand parabolic equations of second order

As a branch of regularity, Schauder estimate plays a very important role in proving existenceand uniqueness for the nonlinear elliptic and parabolic equations. In recent years, there are a number of works concerning Schauder estimate ofanisotropic equations. In this review, we aim to introduce some results about anisotropic Schauder estimate for elliptic and parabolic equations.

Greene Approximation Wide-Angle Parabolic Equation for Radio Propagation

This paper presents the Greene second-order parabolic equation (PE) solution to model radio propagation over irregular terrain. This solution is implemented by finite-difference and shift-map technique. The second-order finite-difference approach performs well for slopes up to about 15°, and discontinuous slope changes up to about 30°, which is better than can be achieved using the split-step/Fourier approach. The results prove that the Greene PE solution has greater propagation angles than does the Claerbout PE solution. Thus, the Greene second-order solution can certainly give better results for large angle propagation over complicated terrain boundaries. Besides, we treat the important problem of accuracy of the different approximations of the PE-based propagation models, and derive the general PE solutions with respect to the terrain slope for the different approximations. It is useful to have collected in one paper, the most important PE approximations for radio propagation.

On boundary and initial values of solutions of a second-order parabolic equation that degenerate on the domain boundary

Properties of the solutions of a parabolic equation in the case of a Keldysh-type degeneracy on the boundary of the domain are investigated. The unique solvability of the first mixed problem for this equation is studied. Necessary and sufficient conditions for the existence of limits of the solution on the lateral surface of a cylindrical domain and on its lower base in L2-type spaces are found.

On exact solutions of a heat-wave type with logarithmic front for the porous medium equation

The paper deals with a nonlinear second-order parabolic equation with partial derivatives, which is usually called “the porous medium equation”. It describes the processes of heat and mass transfer as well as filtration of liquids and gases in porous media. In addition, it is used for mathematical modeling of growth and migration of population. Usually this equation is studied numerically like most other nonlinear equations of mathematical physics. So, the construction of exact solution in an explicit form is important to verify the numerical algorithms. The authors deal with a special solutions which are usually called “heat waves”. A new class of heat-wave type solutions of one-dimensional (plane-symmetric) porous medium equation is proposed and analyzed. A logarithmic heat wave front is studied in details. Considered equation has a singularity at the heat wave front, because the factor of the highest (second) derivative vanishes. The construction of these exact solutions reduces to the integration of a nonlinear second-order ordinary differential equation (ODE). Moreover, the Cauchy conditions lead us to the fact that this equation has a singularity at the initial point. In other words, the ODE inherits the singularity of the original problem. The qualitative analysis of the solutions of the ODE is carried out. The obtained results are interpreted from the point of view of the corresponding heat waves’ behavior. The most interesting is a damped solitary wave, the length of which is constant, and the amplitude decreases.

A fast immersed interface method for the Cahn–Hilliard equation with arbitrary boundary conditions in complex domains

A fast immersed boundary method for the Cahn–Hilliard equation is introduced. The decomposition of the fourth-order non-linear Cahn–Hilliard equation into a system of linear parabolic second-order equations allows to pose arbitrary Neumann or surface wetting conditions on the boundary. In space a finite-volume discretization on a regular Cartesian voxel grid allows the use of fast parabolic solvers via Fourier transform of arbitrary convergence order. For the time discretization, a second-order Runge–Kutta scheme is applied. The polynomial approximation of the chemical potential results in a numerical scheme that is unconditionally gradient-stable and allows large time steps. With an additional pre-conditioner for the linear system, the condition of linear system is minimized. By this the convergence is independent of both spatial discretization and time step size. This allows for the simulation of phase separation in large porous complex domains with three dimensions and over hundred million degrees of freedom while applying arbitrary boundary conditions and using large time steps.

Vector domain decomposition schemes for parabolic equations

A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.

A Note about the Black-Scholes Option Pricing Model under Time-Varying Conditions

By solving the second-order parabolic partial differential equation, the BS option pricing formula can be obtained; The BS pricing model under time-varying conditions is hard to get an explicit solution; We found that there is a same mistake in Zhige Ren’s paper and Xiaochen Han’s paper, and trying to put forward an idea to solve a sort of second-order parabolic partial differential equations.

Viscosity solutions to a model for solid–solid phase transitions driven by material forces

A new phase-field model is formulated to describe martensitic phase transitions driven by material forces, in solid materials, e.g., shape memory alloys. This model is a nonlinear degenerate parabolic equation of second order and its principal part is not in divergence form in multi-dimensional case. The global existence of viscosity solutions to an initial–boundary value problem for this model is proved.

Sparse Beltrami Coefficients, Integral Means of Conformal Mappings and the Feynman-Kac Formula

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula with a sparsely supported potential and introduce a dictionary between the two settings.

Spectral Shift of Cylindrical Heat Equations by Full-State Boundary Feedback

This contribution introduces a full-state boundary feedback for cylindrical heat conductors. The desired decay properties are imposed to the spectrum of linear, second-order parabolic partial differential equations (PDEs). The feedback-scheme exploits periodicity in angular direction for cylindrical coordinates to reduce the 3-dimensional (3-D) system to a set of 2-D subsystems by means of a Fourier transformation. A backstepping-based controller for shifting the spectra for each of the decoupled systems on 2-D domains is presented. It can be shown, that any decay rate of the overall tracking error can be achieved by superposing solely a finite number of these controllers. The validity of the approach is shown by simulations.