Research Article
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Year 2018, Volume: 47 Issue: 6, 1503 - 1511, 12.12.2018

Ebru Ozbilge Ali Demir

Abstract

References

  • Amanov, D., Ashyralyev, A., Initial-boundary value problem for fractional partial differential equations of higher order, Abstr. Appl. Anal. 2012, Article Id: 9733102, 470-484, 2012.
  • Ashyralyev, A., A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl. 357, 232-236, 2009.
  • Ashyralyev, A., Artykov, M., Cakir, Z., A note on fractional parabolic differential and difference equations, AIP Conf. Proceedings 1611, 251-254, 2014.
  • Ashyralyev, A., Cakir, Z., FDM for fractional parabolic equations with the Neumann condition, Adv. Differential Equations, 120, Doi:10.1186/1687-1847-2013-120, 2013.
  • Ashyralyev, A., Dal, F., Finite difference and iteration methods for fractional hyperbolic partial differential equations with neumann condition, Discrete Dyn. Nat. Soc. 2012, Article Id: 434976, 2012.
  • Ashyralyev, A., Hicdurmaz,B., On the numerical solution of fractional schrodinger differential equations with dirichlet condition, Int. J. Comput. Math., 89, 1927-1936, 2012.
  • Ashyralyev, A., Sharifov,Y.A., Existence and Uniqueness of solutions for the system of nonlinear fractional differential equations with nonlocal and integral boundary conditions, Abst. and Appl.Analys., 2012, Article Id: 594802,2012.
  • Baglan, I., Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition, Inv. Probl.in Sci.and Eng., \textbf23 (5), 884-900, 2015.
  • Baglan, I., Kanca, F., Weak generalized and numerical solution for a quasilinear pseudo-parabolic equation with nonlocal boundary condition, Adv. Differential Equations, 2014 (277), Doi:10.1186/1687-1847-2014-277, 2014.
  • Baglan, I., Kanca, F., An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition, Math. Meths. in the Appl. Sci., 38 (5), 851-867, 2015.
  • Cannon, J.R., Lin, Y. An inverse problem fo finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl. 145 (1), 470-484, 1990.
  • Cannon, J. R., Lin, Y. Determination of source parameter in parabolic equations, Mechanica. 27, 85-94, 1992.
  • Cannon, J.R., Lin, Y. Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inv. Prob. 4, 35-44, 1998.
  • Dehghan, M. Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. Diff. Eq. 21, 621-622, 2004.
  • Demir, A., Kanca, F., Ozbilge, E., Numerical solution and distinguishability in time fractional parabolic equation, Bound. Val. Prblms., Article No: 142, 2015.
  • Djrbashian, M. M., Differential Operators of fractional order and boundary value problems in the complex domain, The Gohberg Anniversary Collection, 153-172, Springer, 1989.
  • Erdogan, A. S., Ashyralyev, A. On the second order implicit difference schemes for a right hand side identification problem, Appl.Math.Comp. 226, 212-229, 2014.
  • Erdogan, A. S., Sazaklioglu, A.U., A note on the numerical solution of an identification problem for observing two-phase flow in capillaries, Math. Methds. Appl. Sci., 37, 2393-2405, 2014.
  • Fatullayev, A. F., Numerical procedure for the simultaneous determination of unknown coefficients in a parabolic equation, Appl. Math. Comp. 164, 697-705, 2005.
  • Francesco, M., Luchko, Y., Pagnini, G. The fundamental solution of the space-time fractional diffusion equation, Arxiv prep. cond-mat, p.0702419, 2007.
  • Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics, 29, 129-143, 2002.
  • Luchko, Y., Initial boundary value problems for the one dimentional time-fractional diffusion equation, Frac. Calc. Appl. Analy., 15, 141-160, 2012.
  • Ozbilge, E., Demir,A., Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions, Bound.Value. Prblms., Article No:134, 2014.
  • Ozbilge, E., Demir,A., Identification of unknown coefficient in time fractional parabolic equation with mixed boundary conditions via semigroup approach, Dyn. Syst. Appl., 24(3), 341-348, 2015.
  • Ozbilge, E., Demir,A., Inverse problem for a time-fractional parabolic equation, Jour. of Ineq. and Appl., Article No:81, 2015.
  • Ozbilge, E., Demir, A., Kanca, F., Ozbilge, E., Determination of the unknown source function in time fractional parabolic equation with dirichlet boundary conditions, Appl. Math. Inf. Sci.,10(1), 283-289, 2016.
  • Plociniczak, L., Approximation of the erdelyi-kober operator with application to the time-fractional porous medium equation, SIAM J. Appl. Math., 74(4), 1219-1237, 2014.
  • Plociniczak, L., Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Comm. Nonlin.Sci. Num.Simul., 24(1), 169-183, 2015.
  • Wei, H., Chen, W., Sun, H., Li, X., A coupled method for inverse source problem of spatial fractional anomalous diffusion equations, Inv. Prblms. in Sci. Eng., 18(7), 945-956, 2010.
  • Wei, T., Zhang, Z.Q., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Analys.Boundry. Elemts., 37(1), 23-31, 2013.
  • Zhang, Y., Xiang, X., Inverse source problem for a fractional diffusion equation, Inv. Prblms., 27(3):035010, 2011.

Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition

Year 2018, Volume: 47 Issue: 6, 1503 - 1511, 12.12.2018

Ebru Ozbilge Ali Demir

Abstract

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 $D_{t}^{\alpha }u(x,t)=(k(x)u_{x})_{x}+F(x,t) \quad 0<\alpha \leq 1$, with Robin boundary conditions $u(0,t)=\psi _{0}(t)$, $u_{x}(1,t)=\gamma(u(1,t)-\psi _{1}(t))$. By defining the input-output mappings $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot ]:\mathcal{K}\rightarrow C[0,T]$ 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 $\Phi[\cdot ]$ and $\Psi [\cdot ]$. 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 $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot]:\mathcal{K}\rightarrow C[0,T]$ can be described explicitly.

Keywords

Inverse problem time-fractional parabolic equation distinguishability

References

  • Amanov, D., Ashyralyev, A., Initial-boundary value problem for fractional partial differential equations of higher order, Abstr. Appl. Anal. 2012, Article Id: 9733102, 470-484, 2012.
  • Ashyralyev, A., A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl. 357, 232-236, 2009.
  • Ashyralyev, A., Artykov, M., Cakir, Z., A note on fractional parabolic differential and difference equations, AIP Conf. Proceedings 1611, 251-254, 2014.
  • Ashyralyev, A., Cakir, Z., FDM for fractional parabolic equations with the Neumann condition, Adv. Differential Equations, 120, Doi:10.1186/1687-1847-2013-120, 2013.
  • Ashyralyev, A., Dal, F., Finite difference and iteration methods for fractional hyperbolic partial differential equations with neumann condition, Discrete Dyn. Nat. Soc. 2012, Article Id: 434976, 2012.
  • Ashyralyev, A., Hicdurmaz,B., On the numerical solution of fractional schrodinger differential equations with dirichlet condition, Int. J. Comput. Math., 89, 1927-1936, 2012.
  • Ashyralyev, A., Sharifov,Y.A., Existence and Uniqueness of solutions for the system of nonlinear fractional differential equations with nonlocal and integral boundary conditions, Abst. and Appl.Analys., 2012, Article Id: 594802,2012.
  • Baglan, I., Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition, Inv. Probl.in Sci.and Eng., \textbf23 (5), 884-900, 2015.
  • Baglan, I., Kanca, F., Weak generalized and numerical solution for a quasilinear pseudo-parabolic equation with nonlocal boundary condition, Adv. Differential Equations, 2014 (277), Doi:10.1186/1687-1847-2014-277, 2014.
  • Baglan, I., Kanca, F., An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition, Math. Meths. in the Appl. Sci., 38 (5), 851-867, 2015.
  • Cannon, J.R., Lin, Y. An inverse problem fo finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl. 145 (1), 470-484, 1990.
  • Cannon, J. R., Lin, Y. Determination of source parameter in parabolic equations, Mechanica. 27, 85-94, 1992.
  • Cannon, J.R., Lin, Y. Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inv. Prob. 4, 35-44, 1998.
  • Dehghan, M. Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. Diff. Eq. 21, 621-622, 2004.
  • Demir, A., Kanca, F., Ozbilge, E., Numerical solution and distinguishability in time fractional parabolic equation, Bound. Val. Prblms., Article No: 142, 2015.
  • Djrbashian, M. M., Differential Operators of fractional order and boundary value problems in the complex domain, The Gohberg Anniversary Collection, 153-172, Springer, 1989.
  • Erdogan, A. S., Ashyralyev, A. On the second order implicit difference schemes for a right hand side identification problem, Appl.Math.Comp. 226, 212-229, 2014.
  • Erdogan, A. S., Sazaklioglu, A.U., A note on the numerical solution of an identification problem for observing two-phase flow in capillaries, Math. Methds. Appl. Sci., 37, 2393-2405, 2014.
  • Fatullayev, A. F., Numerical procedure for the simultaneous determination of unknown coefficients in a parabolic equation, Appl. Math. Comp. 164, 697-705, 2005.
  • Francesco, M., Luchko, Y., Pagnini, G. The fundamental solution of the space-time fractional diffusion equation, Arxiv prep. cond-mat, p.0702419, 2007.
  • Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics, 29, 129-143, 2002.
  • Luchko, Y., Initial boundary value problems for the one dimentional time-fractional diffusion equation, Frac. Calc. Appl. Analy., 15, 141-160, 2012.
  • Ozbilge, E., Demir,A., Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions, Bound.Value. Prblms., Article No:134, 2014.
  • Ozbilge, E., Demir,A., Identification of unknown coefficient in time fractional parabolic equation with mixed boundary conditions via semigroup approach, Dyn. Syst. Appl., 24(3), 341-348, 2015.
  • Ozbilge, E., Demir,A., Inverse problem for a time-fractional parabolic equation, Jour. of Ineq. and Appl., Article No:81, 2015.
  • Ozbilge, E., Demir, A., Kanca, F., Ozbilge, E., Determination of the unknown source function in time fractional parabolic equation with dirichlet boundary conditions, Appl. Math. Inf. Sci.,10(1), 283-289, 2016.
  • Plociniczak, L., Approximation of the erdelyi-kober operator with application to the time-fractional porous medium equation, SIAM J. Appl. Math., 74(4), 1219-1237, 2014.
  • Plociniczak, L., Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Comm. Nonlin.Sci. Num.Simul., 24(1), 169-183, 2015.
  • Wei, H., Chen, W., Sun, H., Li, X., A coupled method for inverse source problem of spatial fractional anomalous diffusion equations, Inv. Prblms. in Sci. Eng., 18(7), 945-956, 2010.
  • Wei, T., Zhang, Z.Q., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Analys.Boundry. Elemts., 37(1), 23-31, 2013.
  • Zhang, Y., Xiang, X., Inverse source problem for a fractional diffusion equation, Inv. Prblms., 27(3):035010, 2011.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ebru Ozbilge This is me

Ali Demir

Publication Date December 12, 2018
Published in Issue Year 2018 Volume: 47 Issue: 6

Cite

APA Ozbilge, E., & Demir, A. (2018). Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition. Hacettepe Journal of Mathematics and Statistics, 47(6), 1503-1511.
AMA Ozbilge E, Demir A. Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition. Hacettepe Journal of Mathematics and Statistics. December 2018;47(6):1503-1511.
Chicago Ozbilge, Ebru, and Ali Demir. “Distinguishability of a Source Function in a Time Fractional Inhomogeneous Parabolic Equation With Robin Boundary Condition”. Hacettepe Journal of Mathematics and Statistics 47, no. 6 (December 2018): 1503-11.
EndNote Ozbilge E, Demir A (December 1, 2018) Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition. Hacettepe Journal of Mathematics and Statistics 47 6 1503–1511.
IEEE E. Ozbilge and A. Demir, “Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, pp. 1503–1511, 2018.
ISNAD Ozbilge, Ebru - Demir, Ali. “Distinguishability of a Source Function in a Time Fractional Inhomogeneous Parabolic Equation With Robin Boundary Condition”. Hacettepe Journal of Mathematics and Statistics 47/6 (December 2018), 1503-1511.
JAMA Ozbilge E, Demir A. Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition. Hacettepe Journal of Mathematics and Statistics. 2018;47:1503–1511.
MLA Ozbilge, Ebru and Ali Demir. “Distinguishability of a Source Function in a Time Fractional Inhomogeneous Parabolic Equation With Robin Boundary Condition”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, 2018, pp. 1503-11.
Vancouver Ozbilge E, Demir A. Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition. Hacettepe Journal of Mathematics and Statistics. 2018;47(6):1503-11.

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