Stable reconstruction of the initial condition in parabolic equations from boundary observations

2020 ◽  
Vol 79 (12) ◽  
pp. 3570-3587
Author(s):  
Dinh Nho Hào ◽  
Phan Xuan Thanh ◽  
Bandar Bin-Mohsin ◽  
Nguyen Huu Cong
Keyword(s):  
Parabolic Equations ◽  
Initial Condition ◽  
Stable Reconstruction

scholarly journals A problem of coefficient determination in parabolic equations solved as moment problem

2017 ◽  
Vol 6 (4) ◽  
pp. 109
Author(s):  
Maria Beatriz Pintarelli
Keyword(s):  
Boundary Conditions ◽  
Real Number ◽  
Moment Problem ◽  
Parabolic Equations ◽  
Arbitrary Real Number ◽  
Initial Condition ◽  
Moments Problem ◽  
Inverse Moment

The problem is to find a(t) y w(x; t) such that wt = a(t) (wx)x+r(x; t) under the initial condition w(x; 0) =fi(x) and the boundary conditions w(0; t) = 0 ; wx(0; t) = wx(1; t)+alfa w(1; t) about a region D ={(x; t); 0 <x < 1; t >0}. In addition it must be fulfilled the integral of w (x, t) with respect to x is equal to E(t) where fi(x) , r(x; t) and E(t) are known functions and alfa is an arbitrary real number other than zero.The objective is to solve the problem as an application of the inverse moment problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. In addition, the method is illustrated with several examples.

Determination of the initial condition in parabolic equations from boundary observations

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Dinh Nho Hào ◽  
Nguyen Thi Ngoc Oanh
Keyword(s):  
Variational Problem ◽  
Parabolic Equations ◽  
Numerical Scheme ◽  
Singular Values ◽  
Adjoint Problem ◽  
Splitting Methods ◽  
Initial Condition ◽  
Numerical Examples ◽  
Gradient Of The Functional

AbstractThe problem of determining the initial condition in parabolic equations from boundary observations is studied. It is reformulated as a variational problem and then a formula for the gradient of the functional to be minimized is derived via an adjoint problem. The variational problem is discretized by finite difference splitting methods and solved by the conjugate gradient method. Some numerical examples are presented to show the efficiency of the method. Also as a by-product of the variational method, we propose a numerical scheme for numerically estimating singular values of the solution operator in the inverse problem.

scholarly journals Error estimates for the semidiscrete finite element approximation of linear nonlocal parabolic equations

1992 ◽  
Vol 5 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Dennis E. Jackson
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Finite Element Approximation ◽  
Parabolic Partial Differential Equations ◽  
Element Approximation ◽  
Initial Condition ◽  
Initial Value ◽  
Semidiscrete Approximation ◽  
Nonlocal Parabolic Equations

Existence and uniqueness are proved for nonlocal (in time) for solutions of linear parabolic partial differential equations. Instead of an initial condition, there is a relation connecting the initial value to values of the solution at other times. L2 error estimates are obtained for the semidiscrete approximation of the problem using finite elements in the space variables.

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