A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Distortion-suppression optimization for a class of nonlinear optical systems with feedback

We consider the initial boundary-value problem for the quasilinear diffusion equation ∂ _t u+u − DΔu = K(x, y)(1 + γ(u + ϕ(x, y))) describing the dynamics of optical systems with controlled feedback wave intensity modulation K(x, y) in the presence of incoming-wave phase perturbations ϕ(x, y). The control problem for the parameter K(x, y) is formulated with the objective of smoothing out the spatial nonhomogeneities of the total output phase u(x, y, T) + ϕ(x, y). We prove existence and uniqueness theorems for the generalized solutions of the direct and conjugate problems, solvability theorems for the optimization problems, and Frechet-differentiability of the objective functional. A formula for the functional gradient is derived and the efficiency of the gradient projection method is demonstrated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 80–99, 2005.     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

Blow Up in a Nonlinearly Damped Wave Equation

In this paper we consider the nonlinearly damped semilinear wave equation u_tt – Δu + au_t |u_t|^{m – 2} = bu|u|^{p – 2} associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2].     

Fourth-order finite difference method for 2D parabolic partial differential equations with nonlinear first-derivative terms

We attempt to obtain a two-level implicit finite difference scheme using nine spatial grid points of O(k² + kh² + h⁴) for solving the 2D nonlinear parabolic partial differential equation v₁u_xx + v₂u_yy = f(x, y, t, u, u_x, u_y, u₁) where v₁ and v₂ are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion-convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

A symmetry classification for a class of (2+1)-nonlinear wave equation

In this paper, a symmetry classification of a (2+1)-nonlinear wave equation u_tt−f(u)(u_xx+u_yy)=0 where f(u) is a smooth function on u, using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this (2+1)-nonlinear wave equation.     

Stability and localization of unbounded solutions of a nonlinear heat equation in a plane

The article investigates unbounded solutions of the equation u _t = div (u ^σgrad u) + u ^β in a plane. We numerically analyze the stability of two-dimensional self-similar solutions (structures) that increase with blowup. We confirm structural stability of the simple structure with a single maximum and metastability of complex structures. We prove structural stability of the radially symmetrical structure with a zero region at the center and investigate its attraction region. We study the effect of various perturbations of the initial function on the evolution of self-similar solutions. We further investigate how arbitrary compact-support initial distributions attain the self-similar mode, including distributions whose support is different from a disk. We show that the self-similar mode described by a simple radially symmetrical structure is achieved only in the central region, while the entire localization region does not have enough time to transform into a disk during blowup. We show for the first time that simple structures may merge into a complex structure, which evolves for a long time according to self-similar law.     

A new highly accurate discretization for three‐dimensional singularly perturbed nonlinear elliptic partial differential equations

We employ a new fourth‐order compact finite difference formula based on arithmetic average discretization to solve the three‐dimensional nonlinear singularly perturbed elliptic partial differential equation ε(u_xx + u_yy + u_zz) = f(x, y, z, u, u_x, u_y, u_z), 0 < x, y, z < 1, subject to appropriate Dirichlet boundary conditions prescribed on the boundary, where ε > 0 is a small parameter. We also describe new fourth‐order methods for the estimates of (?u/?x), (?u/?y), and (?u/?z), which are quite often of interest in many physical problems. In all cases, we require only a single computational cell with 19 grid points. The proposed methods are directly applicable to solve singular problems without any modification. We solve three test problems numerically to validate the proposed derived fourth‐order methods. We compare the advantages and implementation of the proposed methods with the standard central difference approximations in the context of basic iterative methods. Numerical examples are given to verify the fourth‐order convergence rate of the methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Computability of Solutions of the Korteweg‐de Vries Equation

In this paper we study computability of the solutions of the Korteweg‐de Vries (KdV) equation u_t + uu_x + u_xxx = 0. This is one of the open problems posted by Pour‐El and Richards [25]. Based on Bourgain's new approach to the initial value problem for the KdV equation in the periodic case, we show that the periodic solution u (x, t) of the KdV equation is computable if the initial data is computable.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

Optimal Systems and Group Classification of (1+2)-Dimensional Heat Equation

The optimal systems and symmetry breaking interactions for the (1+2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H ₂. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H ₂. A list of representatives of all Lie subalgebras of the Lie algebra h ₂ of the Lie group H ₂ is given in the form of tables and many of their properties are established. We derive the most general interactions F(t,x,y,u,u _x ,u _y ) such that the equation u _t =u _xx +u _yy +F(t,x,y,u,u _x ,u _y ) is invariant under each subgroup.     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation u_ty = u_xu_xy ? u_yu_xx, (2) the Pavlov equation u_yy = u_tx + u_yu_xx ? u_xu_xy, (3) the universal hierarchy equation u_yy = u_tu_xy ? u_yu_tx, and (4) the modified Veronese web equation u_ty = u_tu_xy ? u_yu_tx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.     

Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.