Publication Date:
2015-08-17
Description:
This article deals with the mathematical analysis of the inverse problem of identifying the distinguishability of input-output mappings in the linear time fractional inhomogeneous parabolic equation D t α u ( x , t ) = ( k ( x ) u x ) x + r ( t ) F ( x , t ) , 0 〈 α ≤ 1 , with mixed boundary conditions u ( 0 , t ) = ψ 0 ( t ) , u x ( 1 , t ) = ψ 1 ( t ) . By defining the input-output mappings Φ [ ⋅ ] : K → C 1 [ 0 , T ] and Ψ [ ⋅ ] : K → C [ 0 , T ] the inverse problem is reduced to the problem of their invertibility. Hence, the main purpose of this study is to investigate the distinguishability of the input-output mappings Φ [ ⋅ ] and Ψ [ ⋅ ] . Moreover, the measured output data f ( t ) and h ( t ) can be determined analytically by a series representation, which implies that the input-output mappings Φ [ ⋅ ] : K → C 1 [ 0 , T ] and Ψ [ ⋅ ] : K → C [ 0 , T ] can be described explicitly, where Φ [ r ] = k ( x ) u x ( x , t ; r ) | x = 0 and Ψ [ r ] = u ( x , t ; r ) | x = 1 . Also, numerical tests using finite difference scheme combined with an iterative method are presented.
Print ISSN:
1687-2762
Electronic ISSN:
1687-2770
Topics:
Mathematics
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