Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

Abstract

The reconstruction of an unknown solely time-dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time-discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time-discrete approximation scheme in space. Finally, full-discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment.

Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

“…An inverse source problem (ISP) of determining a missing solely time-dependent function in a nonlinear parabolic integrodifferential setting has been studied in [17]. This covers the inverse source problem for determination of unknown h(t) if the source is in a separated form f (x)h(t) as in the classical relation (10).…”

Recovery of the time‐dependent zero‐order coefficient in a fourth‐order parabolic problem

“…Using (3.3) and the functions in (3.10) and (3.11), for all $$$ t\in \left(0,T\right] $$$ and any $$$ \chi \in {H}^2\left(0,1\right)\cap {H}_0^1\left(0,1\right) $$$, we have $$$$ {\int}_0^1{\partial}_t{u}^K(t)\chi dx+{\int}_0^1{\partial}_x^2{\overline{u}}^K(t){\partial}_x^2\chi dx+{\overline{q}}^K(t){\int}_0^1{\overline{u}}^K(t)\chi dx={\int}_0^1{\overline{f}}^K(t)\chi dx. $$$$ Inspired by the work [17], Rothe's method can be utilized to prove the existence and uniqueness of the solution to the inverse problem (1.1) and (1.2) according to the estimates in Lemma 3.2, we illustrate the result without proof as follows.Theorem Let the assumptions of Lemma 3.2 hold , then there exists a unique solution …”

“…For the numerical reconstruction of the unknown time‐dependent quantity $$$ q(t) $$$ of (1.1) and (1.2), one method is to solve the equivalent inverse source problem obtained from the transformations directly by using the finite‐difference scheme, and the cubic spline function method [16]. Another numerical method applied in this work is the time‐discrete method, which is applied in second‐order parabolic problem [17] and fourth‐order inverse hyperbolic problem [18], the convergence error estimates for such scheme is obtained rigorous for the noisy measurement. In addition, the predictor–corrector method is introduced to improve the accuracy of the unknown quantity.…”

The reconstruction of a solely time-dependent load in a simply supported non-homogeneous Euler–Bernoulli beam

“…This technique starts with a transformation of the variational formulation of the inverse problem with unknowns u and h into a variational problem in which the unknown h is written in terms of the unknown u. This approach has already been successfully applied to inverse problems containing parabolic or hyperbolic second-order partial differential equations that have been studied for example in [9][10][11][12][13][14]. According to our knowledge, it is the first time that this technique is used for fourth-order operators.…”

“…It is important to note that the fourth-order differential operator in (1) makes the analysis definitely more complex and challenging, even in the one-dimensional case considered herein. The numerical schemes developed in [9][10][11][12][13][14] are based on the following additional integral-type measurement…”