Ritz–Galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition

References

• Cannon JR, Lin Y. An inverse problem of finding a parameter in a semilinear heat equation. J Math Anal Appl. 1990;145:470–484.
   Web of Science ®Google Scholar
• Chen Q, Liu JJ. Solving an inverse parabolic problem by optimization from final measurement data. J Comput Appl Math. 2006;193:183–203.
   Web of Science ®Google Scholar
• Choulli M, Yamamoto M. An inverse parabolic problem with non-zero initial condition. Inverse Probl. 1997;13:19–27.
   Web of Science ®Google Scholar
• Dehghan M. Determination of a control parameter in the two-dimensional diffusion equation. Appl Numer Math. 2001;37:489–502.
   Web of Science ®Google Scholar
• Dehghan M. Finding a control parameter in one-dimensional parabolic equation. Appl Math Comput. 2003;135:491–503.
   Web of Science ®Google Scholar
• Dehghan M. Parameter determination in a partial differential equation from the overspecified data. Math Comput Model. 2005;41:196–213.
   Web of Science ®Google Scholar
• Dehghan M. Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement. Numer Methods Partial Differ Equ. 2005;21:611–622.
   Web of Science ®Google Scholar
• Dehghan M, Tatari M. Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math Comput Model. 2006;44:1160–1168.
   Web of Science ®Google Scholar
• Rundell W. The determination of a parabolic equation from initial and final data. Proc Amer Math Soc. 1987;99:637–642.
   Web of Science ®Google Scholar
• Yamamoto M, Zou J. Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Probl. 2001;17:1181–1202.
   Web of Science ®Google Scholar
• Yang L, Yu JN, Deng ZC. An inverse problem of identifying the coefficient of parabolic equation. Appl Math Model. 2008;32(10):1984–1995.
   Web of Science ®Google Scholar
• Yousefi SA. Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse Probl Sci Eng. 2009;17(6):821–828.
   Web of Science ®Google Scholar
• Li H, Zhou JR. Direct and inverse problem for the parabolic equation with initial value and time-dependent boundaries. Appl Anal. 2016;95(6):1307–1326.
   Web of Science ®Google Scholar
• Franks SJ, Byrne HM, Mudhar HS, et al. Mathematical modelling of comedo ductal carcinoma in situ of the breast. Math Med Biol. 2003;20:277–308.
   PubMed Web of Science ®Google Scholar
• Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumours. J Math Biol. 1999;38:262–284.
   PubMed Web of Science ®Google Scholar
• Greenspan HP. Models for the growth of tumour by diffusion. Stud Appl Math. 1972;52:317–340.
   Google Scholar
• Li H, Xu Y, Zhou JR. A free boundary problem arising from DCIS mathematical model. Math Methods Appl Sci. 2017;40(10):3566–3579.
   Web of Science ®Google Scholar
• Xu Y, Gilbert R. Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Math Comput Model. 2009;49:814–828.
   Web of Science ®Google Scholar
• Xu Y. A free boundary problem model of ductal carcinoma in situ. Discrete Contin Dyn Syst-Ser B. 2004;4(1):337–348.
   Web of Science ®Google Scholar
• Xu Y. A mathematical model of ductal carcinoma in situ and its characteristic stationary solutions. In: Begehr H, Gilbert R, Muldoon M, et al., editors. Advances in analysis. Singapore: World Scientific Publisher; 2005. p. 365–374.
   Google Scholar
• Xu Y. A free boundary problem of diffusion equation with integral condition. Appl Anal. 2006;85(9):1143–1152.
   Google Scholar
• Xu Y. A free boundary problem of parabolic complex equation. Complex Var Elliptic Equ. 2006;51(8–11):945–951.
   Google Scholar
• Xu Y. An inverse problem for the free boundary model of ductal carcinoma in situ. In: Begehr H, Nicolosi F, editors. More progresses in analysis. Singapore: World Science Publisher; 2008. p. 1429–1438.
   Google Scholar
• Dehghan M, Yousefi SA, Rashedi K. Ritz--Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions. Inverse Probl Sci Eng. 2013;21:500–523.
   Web of Science ®Google Scholar
• Rashedi K, Adibi H, Dehghan M. Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation. Comput Math Appl. 2013;65:1990–2008.
   Web of Science ®Google Scholar
• Bouziani A, Merazga N, Benamira S. Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions. Nonlinear Anal: Theory Methods Appl. 2008;69:1515–1524.
   Web of Science ®Google Scholar
• Zhou JR, Li H, Xu Y. Ritz--Galerkin method for solving a parabolic equation with non-local and time-dependent boundary conditions. Math Methods Appl Sci. 2016;39:1241–1253.
   Web of Science ®Google Scholar
• Yousefi SA, Barikbin Z. Ritz--Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non-classic boundary conditions. Int J Numer Methods Heat Fluid Flow. 2012;22:39–48.
   Web of Science ®Google Scholar
• Zhou JR, Li H. A Ritz--Galerkin approximation to the solution of parabolic equation with moving boundaries. Bound Value Probl. 2015;2015:236.
   Google Scholar
• Kress R. Linear integral equations. New York (NY): Spring-Verlag; 1989.
   Google Scholar
• Aminikhah H, Biazar J. A new analytical method for solving systems of Volterra integral equations. Int J Comput Math. 2010;87(5):1142–1157.
   Web of Science ®Google Scholar
• Balakumar V, Murugesan K. Numerical solution of systems of linear Volterra integral equations using block-pulse functions. Malaya J Mat S. 2013;1:77–84.
   Google Scholar
• Linz P. Analytical and numerical methods for Volterra equations. Vol. 7. Philadelphia: SIAM; 1985.
   Google Scholar