Nguyen Van Duc

A non-local boundary value problem method for the Cauchy problem for elliptic equations

Let H be a Hilbert space with norm || ||, A:D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equations is regularized by the well-posed non-local boundary value problem with a ≥ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

Stability results for backward parabolic equations with time-dependent coefficients

Let H be a Hilbert space with the norm || || and A(t)?(0 ? t ? T) be positive self-adjoint unbounded operators from D(A(t))?H to H. In the paper, we establish stability estimates of H?lder type and propose a regularization method for the ill-posed backward parabolic equation with time-dependent coefficients Our stability estimates improve the related results by Krein (1957 Dokl. Akad. Nauk SSSR 114 1162?5), and Agmon and Nirenberg (1963 Commun. Pure Appl. Math. 16 121?239). Our regularization method with a priori and a posteriori parameter choice yields error estimates of H?lder type. This is the only result when a regularization method for backward parabolic equations with time-dependent coefficients provides a convergence rate.

Stability estimates for Burgers-type equations backward in time

Abstract We prove stability estimates of Hölder-type for Burgers-type equations ut = (a(x,t)ux)x - d(x,t)uux + f(x,t), (x,t) ∈ (0,1)×(0,T), u(0,t) = g0(t), u(1,t) = g1(t), 0 ≤ t ≤ T, backward in time, with a(x,t), d(x,t), g0(t), g1(t), f(x,t) being smooth functions, under relatively weak conditions on the solutions.

A non-local boundary value problem method for semi-linear parabolic equations backward in time

The ill-posed semi-linear parabolic equation backward in time with the positive self-adjoint unbounded linear operator A and being given is regularized by the well-posed non-local boundary value problem Under the condition , a priori and a posteriori parameter choice rules are suggested which yield the error estimate for some positive constant C.

An a posteriori mollification method for the heat equation backward in time

Abstract The heat equation backward in time u t = u x ⁢ x , x ∈ ℝ , t ∈ ( 0 , T ) , ∥ u ⁢ ( ⋅ , T ) - φ ⁢ ( ⋅ ) ∥ L p ⁢ ( ℝ ) ⩽ ε , u_{t}=u_{xx},\quad x\in\mathbb{R},\,t\in(0,T\/),\qquad\|u(\,\cdot\,,T\/)-% \varphi(\,\cdot\,)\|_{L_{p}(\mathbb{R})}\leqslant\varepsilon, subject to the constraint ∥ u ⁢ ( ⋅ , 0 ) ∥ L p ⁢ ( ℝ ) ⩽ E {\|u(\,\cdot\,,0)\|_{L_{p}(\mathbb{R})}\leqslant E} , with T > 0 {T>0} , φ ∈ L p ⁢ ( ℝ ) {\varphi\in L_{p}(\mathbb{R})} , 0 < ε < E {0<\varepsilon<E} , 1 < p < ∞ {1<p<\infty} being given, is regularized by the well-posed mollified problem u t ν = u x ⁢ x ν , x ∈ ℝ , t ∈ ( 0 , T ) , u ν ⁢ ( x , T ) = S ν ⁢ ( φ ) ⁢ ( x ) , ν > 0 , u_{t}^{\nu}=u_{xx}^{\nu},\quad x\in\mathbb{R},\,t\in(0,T\/),\qquad u^{\nu}(x,T% \/)=S_{\nu}(\varphi)(x),\quad\nu>0, where S ν ⁢ ( φ ) ⁢ ( x ) = 1 π ⁢ ∫ - ∞ + ∞ sin ⁡ ( ν ⁢ y ) y ⁢ φ ⁢ ( x - y ) ⁢ 𝑑 y . S_{\nu}(\varphi)(x)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\sin(\nu y)}{y}% \varphi(x-y)\,dy. An a posteriori parameter choice rule for this regularization method is suggested, which yields the error estimate ∥ u ⁢ ( ⋅ , t ) - u ν ⁢ ( ⋅ , t ) ∥ L p ⁢ ( ℝ ) ⩽ c ⁢ ε t / T ⁢ E 1 - t / T   for all ⁢ t ∈ [ 0 , T ] . \|u(\,\cdot\,,t)-u^{\nu}(\,\cdot\,,t)\|_{L_{p}(\mathbb{R})}\leqslant c% \varepsilon^{t/T}E^{1-t/T}\quad\text{for all }t\in[0,T]. Furthermore, we establish stability estimates of Hölder type for all derivatives of the solutions with respect to x and t.