An inverse problem of identifying   the time-dependent potential and source terms in a two-dimensional parabolic equation

Mousa J. Huntul; İbrahim Tekin

Makale detayları

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Yazarlar:

Mousa J. Huntul | İbrahim Tekin

Yayıncı:

Hacettepe University

Ana Başlıklar:

Mathematical Sciences

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 6, pp.1578-1599

Belge Tipi:

Makale

Yayın Tarihi:

3/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301859

pISSN:

2651-477X

Anahtar kelimeler:

Özet:

 In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional 2D parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms, we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit ADE method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem IP unique.

 In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional 2D parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms, we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit ADE method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem IP unique.

 In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional 2D parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms, we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit ADE method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem IP unique.

 In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional 2D parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms, we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit ADE method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem IP unique.

Makale detayları

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Yazarlar:

Mousa J. Huntul | İbrahim Tekin

Yayıncı:

Hacettepe University

Ana Başlıklar:

Mathematical Sciences

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 6, pp.1578-1599

Belge Tipi:

Makale

Yayın Tarihi:

3/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301859

pISSN:

2651-477X

Existence and uniqueness

Contraction mapping

2D parabolic equation

Inverse identification problem

Nonlinear optimization

Tikhonov regularization

Sayı

Makale detayları

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Yazarlar: Mousa J. Huntul | İbrahim Tekin

Yayıncı: Hacettepe University

Ana Başlıklar: Mathematical Sciences

Dergi: Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 6, pp.1578-1599

Belge Tipi: Makale

Yayın Tarihi: 3/2023

Belge Dili: İngilizce

Document ID (Org No.): 2001301859

pISSN: 2651-477X

Existence and uniqueness

Contraction mapping

2D parabolic equation

Inverse identification problem

Nonlinear optimization

Tikhonov regularization

Sayı

Yazarlar:

Mousa J. Huntul | İbrahim Tekin

Yayıncı:

Hacettepe University

Ana Başlıklar:

Mathematical Sciences

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 6, pp.1578-1599

Belge Tipi:

Makale

Yayın Tarihi:

3/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301859

pISSN:

2651-477X