Identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation by semigroup approach with mixed boundary conditions

In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k(u(x, t)) in the quasi‐linear parabolic equation u_t(x, t)=(k(u(x, t))u_x(x, t))_x, with Dirichlet boundary conditions u_x(0, t)=ψ₀, u(1, t)=ψ₁. The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : ??→C¹[0, T], Ψ[·] : ??→C¹[0, T] using semigroup theory. In this article, it is shown that if the null space of semigroups T(t) and S(t) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis for the identification of an unknown diffusion coefficient via semigroup approach

This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the inhomogenenous quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C¹[0, T], Ψ[·]:??→C¹[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient

This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C¹[0, T], Ψ[·]: ?? → C¹[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

A Levenberg–Marquardt Scheme for Nonlinear Image Registration

This paper presents a Levenberg—Marquardt scheme to obtain a displacement vector field u(x)=(u ₁(x),u ₂(x)) ^t , which matches two images recorded with the same imaging machinery. The displacement vector should transform the image location x=(x ₁,x ₂) ^t of an image T, such that the grey level are equal to another image R. The so-called mono-modal image registration problem leads to minimize the nonlinear least squares functional D(u(x))=R(x)–T(x–u(x))².To apply the Levenberg—Marquardt method, we replace the nonlinear functional D by its linearization around a current approximation. The resulting quadratic minimization problem is ill-posed, due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. We use an auxiliary Lagrange term borrowed from linear elasticity theory, which incorporates smoothness constraints to the displacement field. Finally, numerical experiments demonstrate the robustness and effectiveness of the proposed approach.This revised version was published online in October 2005 with corrections to the Cover Date.     

Monotonicity of input-output mappings in inverse coefficient and source problems for parabolic equations

This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented.     

Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

The problem of determining the source term F(x, t) in the linear parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the measured data at the final time u(x, T) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µ _T (x) ? u(x, T)‖ ₀ ² can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.     

The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u_t = u_xx + p(t)u + φ, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫₀¹k(x)u(x,t)dx = E(t), 0 ≤ t ≤ T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫₀^s(t) u(x,t)dx = E(t), 0 < t ≤ T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

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.     

On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u⁰₋(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰₋(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰₋ and u⁰₊, a global shock front weak solution u(x, t) = u₋(x, t) for x < ϕ(t), u(x, t) = u₊(x, t) for x > ϕ(t), where u₋ and u₊ are the strong solutions corresponding (respectively) to u⁰₋ and u⁰₊ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u₋(ϕ(t), t) + u₊(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Asymptotic Behaviour of C0–Semigroups with Bounded Local Resolvents

Let {T(t)}_t≥0 be a C₀–semigroup on a Banach space X with generator A, and let H^∞_T be the space of all x ∈ X such that the local resolvent λ ↦ R(λ, A)x has a bounded holomorphic extension to the right half–plane. For the class of integrable functions ϕ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus ϕ ↦ T_ϕ, as operators on H^∞_T. Weshow that each orbit T(·)T_ϕx is bounded and uniformly continuous, and T(t)T_ϕx → 0 weakly as t → ∞, and we give a new proof that ∥T(t)R(μ, A)x∥ = O(t). We also show that ∥T(t)T_ϕx∥ → 0 when T is sun –reflexive, and that ∥T(t)R(μ, A)x∥ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H^∞_T.     

Wave equations with time-dependent spatial operators of higher order

We study the initial-boundary value problem for ?_t²u(t,x)+A(t)u(t,x)+B(t)?_tu(t,x)=f(t,x) on [0,T]×Ω(Ω??ⁿ) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

On a nonlinear stochastic integral equation with application to control systems

The aim of the present paper is to study a nonlinear stochastic integral equation of the form

On a stopped functional for a bidimensional process

Consider a bidimensional process Q_t = (x_t, y_t) consisting of an Ornstein–Uhlenbeck process and a geometric Brownian motion, respectively. Let T be the first time the process x_t hits a constant positive level b > 0. Under certain conditions, we give an explicit form for a stopped functional.u(x,y)=E^x,y[y^m(T)h(x(T))e^-αT],where m, α > 0 are fixed constants and h : R₊ → R is a bounded continuous function.The present result is derived using the method of similarity solutions and the result has many applications in mathematical finance.