A direct numerical procedure for the Cauchy problem for the heat equation

For the Cauchy problem, u_t = u_xx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, u_x(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υ_t = υ_xx, 0 < x, 0 < t ? T, υ_x(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation u_tt + c(t)u_t + b(t)u = a(t)u_xx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.     

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     

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.     

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.     

Generalized periodic solutions of quasilinear equations

We study a boundary-value periodic problem for the quasilinear equationu _ff ?u _xx =F[u,u _f u _x ],u(0,t) =u (π,t),u (x, t + π/q) =u(x, t), 0 ≤x ≤π,t ∈ ?,q ∈ ?. We establish conditions under which the theorem on the uniqueness of a smooth solution is true.     

Analytic-numerical solutions with a priori error bounds for a class of strongly coupled mixed partial differential systems

This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type u_t = Au_xx, u(0,t) + u_x(0,t) = 0, Bu(1,t) + Cu_x(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

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.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Boundary value problems for second-order differential systems

Sufficient conditions for the existence of a solution to the BVP x″ = f(t, x, x′), x(0) = Q₀x(1), x′(0) = Q₁x′(1) are obtained. Here Q₀, Q₁ are nonsingular n × n matrices, with Q₀ orthogonal. These extend results for the periodic case and are proved via a modification of a degree-theoretic approach.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Regularization of a non-characteristic Cauchy problem for the heat equation

The non-characteristic Cauchy problem for the heat equation u_xx(x,t) = u₁(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), u_x(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

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.