### Details

#### Title

An analytical method for solving the two-phase inverse Stefan problem#### Journal title

Bulletin of the Polish Academy of Sciences: Technical Sciences#### Yearbook

2015#### Volume

63#### Numer

No 3#### Publication authors

#### Divisions of PAS

Nauki Techniczne#### Coverage

583-590#### Publisher

Polish Academy of Sciences#### Date

2015[2015.01.01 AD - 2015.12.31 AD]#### Identifier

ISSN 0239-7528, eISSN 2300-1917#### References

Akyildiz (2008), Magnetohydrodynamic flow of a viscoelastic fluid A, Phys Lett, 372, 3380, doi.org/10.1016/j.physleta.2008.01.073 ; Johansson (2011), Numerical approximation of the one - dimensional inverse Cauchy - Stefan problem using a method of fundamental solutions, Inverse Probl Sci Eng, 19, 659, doi.org/10.1080/17415977.2011.579610 ; Araghi (2011), Numerical solution of nonlinear Volterra - Fredholm integro - differential equations using homotopy analysis method, Appl Math Comput, 37, 1. ; Grzymkowski (2006), One - phase inverse Stefan problems solved by Adomian decomposition method, Comput Math Appl, 51, 33, doi.org/10.1016/j.camwa.2005.08.028 ; Słota (2007), Direct and inverse one - phase Stefan problem solved by variational iteration method, Comput Math Appl, 54, 1139, doi.org/10.1016/j.camwa.2006.12.061 ; Słota (2009), Identification of the cooling condition in and D continuous casting processes Heat TransferB, Numer, 55, 155. ; Ren (2007), Application of the heat - balance integral to an inverse Stefan problem, Int J Therm Sci, 46, 118, doi.org/10.1016/j.ijthermalsci.2006.04.013 ; Chauhan (2012), Magnetohydrodynamic slip flow and heat transfer in a porous medium over a stretching cylinder : homotopy analysis method Heat TransferA, Numer, 62, 136. ; Słota (2010), The application of the homotopy perturbation method to one - phase inverse Stefan problem, Int Comm Heat Mass Transf, 37, 587, doi.org/10.1016/j.icheatmasstransfer.2010.03.009 ; Liao (2004), On the homotopy analysis method for nonlinear problems, Appl Math Comput, 147, 499, doi.org/10.1016/S0096-3003(02)00790-7 ; Vosughi (2011), A new analytical technique to solve Volterra s integral equations, Math Methods Appl Sci, 34, 1243, doi.org/10.1002/mma.1436 ; Hetmaniok (2015), Solution of the one - phase inverse Stefan problem by using the homotopy analysis method ( to be published, Appl Math Modelling, doi.org/10.1016/j.apm.2015.02.025 ; Odibat (2010), A study on the convergence of homotopy analysis method, Appl Math Comput, 217, 782, doi.org/10.1016/j.amc.2010.06.017 ; Ghoreishi (2011), Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth - order integro - differential equation, Math Methods Appl Sci, 34, 1833, doi.org/10.1002/mma.1483 ; Hetmaniok (2014), Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind, Numer Algor, 163, doi.org/10.1007/s11075-013-9781-0 ; Brzeziński (2014), High - accuracy numerical integration methods for fractional order derivatives and integrals computations, Bull Pol Tech, 62, 723. ; Liao (2010), An optimal homotopy - analysis approach for strongly nonlinear differential equations, Commun Nonlinear Sci Numer Simulat, 15, 2003, doi.org/10.1016/j.cnsns.2009.09.002 ; Słota (2011), Restoring boundary conditions in the solidification of pure metals & Structures, Comput, 89, 48. ; Johansson (2013), A meshless method for an inverse two - phase one - dimensional linear Stefan problem, Inverse Probl Sci Eng, 21, 17, doi.org/10.1080/17415977.2012.665906 ; Mitkowski (2013), Fractional - order models of the supercapacitors in the form of RC ladder networks, Bull Pol Tech, 61, 581. ; Hetmaniok (2013), Experimental verification of immune recruitment mechanism and clonal selection algorithm applied for solving the inverse problems of pure metal solidification, Int Comm Heat Mass Transf, 47, 7, doi.org/10.1016/j.icheatmasstransfer.2013.07.009 ; Abdulaziz (2010), On convergence of homotopy analysis method and its modification for fractional modified KdV equations, Appl Math Comput, 33, 61. ; Hetmaniok (2012), Determination of optimal parameters for the immune algorithm used for solving inverse heat conduction problems with and without a phase change Heat TransferB, Numer, 62, 462. ; Abbasbandy (2011), A new analytical technique to solve Fredholm s integral equations, Numer Algor, 56, 27, doi.org/10.1007/s11075-010-9372-2 ; Słota (2008), Solving the inverse Stefan design problem using genetic algorithms, Inverse Probl Sci Eng, 16, 829, doi.org/10.1080/17415970801925170 ; Hetmaniok (2014), An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit, Bull Pol Tech, 62, 413. ; Zurigat (2010), The homotopy analysis method for handling systems of fractional differential equations Modelling, Appl Math, 34, 24. ; Słota (2011), Homotopy perturbation method for solving the two - phase inverse Stefan problem Heat TransferA, Numer, 59, 755. ; Liao (1998), Homotopy analysis method : a new analytic method for nonlinear problems, Appl Math Mech Engl Ed, 19, 957, doi.org/10.1007/BF02457955 ; Hristov (2007), An inverse Stefan problem relevant to boilover : heat balance integral solutions and analysis Science, Thermal, 11, 141, doi.org/10.2298/TSCI0702141H ; Liu (2011), Solving two typical inverse Stefan problems by using the Lie - group shooting method Heat Mass Transfer, Int J, 54, 1941. ; Van Gorder (2012), Control of error in the homotopy analysis of semi - linear elliptic boundary value problems, Numer Algor, 61, 613, doi.org/10.1007/s11075-012-9554-1 ; Turkyilmazoglu (2011), Some issues on HPM and HAM : a convergence scheme Modelling, Math Comput, 53, 1929. ; Sowa (2014), A subinterval - based method for circuits with fractional order elements, Bull Pol Tech, 62, 449. ; Marois (2012), What is the most suitable fixed grid solidification method for handling time - varying inverse Stefan problems in high temperature industrial furnaces ? Heat Mass Transfer, Int J, 55, 5471. ; Shidfar (2011), Approximate analytical solutions of the nonlinear reaction - diffusion - convection problems Modelling, Math Comput, 53, 261. ; Yabushita (2007), An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, Phys A Math Theor, 40, 8403, doi.org/10.1088/1751-8113/40/29/015 ; Fan (2013), Optimal homotopy analysis method for nonlinear differential equations in the boundary leyer, Numer Algor, 62, 337, doi.org/10.1007/s11075-012-9587-5 ; Okamoto (2007), A regularization method for the inverse design of solidification processes with natural convection Heat Mass Transfer, Int J, 50, 4409. ; Abbasbandy (2006), Homotopy analysis method for heat radiation equations, Int Comm Heat Mass Transf, 34, 380, doi.org/10.1016/j.icheatmasstransfer.2006.12.001 ; Johansson (2011), A method of fundamental solutions for the one - dimensional inverse Stefan problem Modelling, Appl Math, 35, 4367.#### DOI

10.1515/bpasts-2015-0068