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

References

• Shidfar, A, Caramali, GR, and Damirchi, J, 2006. An inverse heat conduction problem with a nonlinear source term, Nonlinear Anal. 65 (2006), pp. 615–621.
   Web of Science ®Google Scholar
• Cannon, JR, and DuChateau, P, 1998. Structural identification of a term in a heat equation, Inverse Probl. 14 (1998), pp. 535–551.
   Web of Science ®Google Scholar
• Hon, YC, and Wei, T, 2004. A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004), pp. 489–495.
   Web of Science ®Google Scholar
• Berres, S, , Identification of piecewise linear diffusion function in convection–diffusion equation with overspecified boundary, AIP Conf. Proc. 1048 (2008), pp. 76–79.
   Google Scholar
• Cannon, JR, and Dunninger, DR, 1970. Determination of an unknown forcing function in a hyperbolic equation from overspecified data, Ann. Mat. Pure Appl. 85 (4) (1970), pp. 49–62.
   Google Scholar
• Cannon, JR, and DuChateau, P, 1980. An inverse problem for determination an unknown source term in a heat equation, J. Math. Appl. 75 (2) (1980), pp. 465–485.
   Web of Science ®Google Scholar
• Dehghan, M, 2001. An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model. 25 (2001), pp. 743–754.
   Web of Science ®Google Scholar
• Dehghan, M, 2003. Finding a control parameter in one-dimensional parabolic equations, Appl. Math. Comput. 135 (2003), pp. 491–503.
   Web of Science ®Google Scholar
• Dehghan, M, 2005. Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model. 41 (23) (2005), pp. 196–213.
   Web of Science ®Google Scholar
• Dehghan, M, and Shakeri, F, 2009. Method of lines solutions of the parabolic inverse problem with an overspecification at a point, Numerical Algorithms 50 (2009), pp. 417–437.
   Web of Science ®Google Scholar
• Fatullayev, AG, and Cula, S, 2009. An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation, Appl. Math. Lett. 22 (2009), pp. 1033–1037.
   Web of Science ®Google Scholar
• Johansson, BT, and Lesnic, D, 2007. A variational method for identifying a spacewise-dependent heat source, IMA J. Appl. Math. 72 (2007), pp. 748–760.
   Web of Science ®Google Scholar
• Johansson, BT, and Lesnic, D, 2008. A procedure for determining a spacewise dependent heat source and the initial temperature, Appl. Anal. 87 (2008), pp. 265–276.
   Web of Science ®Google Scholar
• Mohebbi, A, and Dehghan, M, 2010. High-order scheme for determination of a control parameter in an inverse problem from the overspecified data, Comput. Phys. Comm. 181 (2010), pp. 1947–1954.
   Web of Science ®Google Scholar
• Rashedi, K, and Yousefi, SA, 2011. Ritz–Galerkin method for solving a class of inverse problems in the parabolic equations, Int. J. Nonlinear Sci. 12 (2011), pp. 498–502.
   Google Scholar
• Saadatmandi, A, and Dehghan, M, 2010. Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications, Int. J. Comput. Math. 87 (2010), pp. 997–1008.
   Web of Science ®Google Scholar
• Slodicka, M, Lesnic, D, and Onyango, TTM, 2010. Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem, Inverse Probl. Sci. Eng. 18 (2010), pp. 65–81.
   Web of Science ®Google Scholar
• Trucu, D, Ingham, DB, and Lesnic, D, 2009. An inverse coefficient identification problem for the bio-heat equation, Inverse Probl. Sci. Eng. 17 (2009), pp. 65–83.
   Web of Science ®Google Scholar
• Yousefi, SA, 2009. Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method, Inverse Probl. Sci. Eng. 17 (2009), pp. 821–828.
   Web of Science ®Google Scholar
• Yousefi, SA, and Dehghan, M, 2009. Legendre multiscaling functions for solving the one-dimensional parabolic inverse problem, Numer. Methods Partial Diff. Eqns. 25 (2009), pp. 1502–1510.
   Web of Science ®Google Scholar
• Lakestani, M, and Dehghan, M, 2010. The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math. 235 (2010), pp. 669–678.
   Web of Science ®Google Scholar
• Cannon, JR, 1984. The One-dimensional Heat Equation. Reading, MA: Addison-Wesley; 1984.
   Google Scholar
• Idrees Bhatti, M, and Bracken, P, 2007. Solution of differential equations in a Bernstein polynomials basis, J. Comput. Appl. Math. 205 (2007), pp. 272–280.
   Web of Science ®Google Scholar
• Rivlin, TJ, 1969. An Introduction to the Approximation of the Functions. New York: Dover; 1969.
   Google Scholar
• Yousefi, SA, Barikbin, Z, and Dehghan, M, 2012. 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 22 (1) (2012), pp. 39–48.
   Web of Science ®Google Scholar
• Yousefi, SA, Behroozifar, M, and Dehghan, M, 2011. The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass, J. Comput. Appl. Math. 235 (2011), pp. 5272–5283.
   Web of Science ®Google Scholar
• Yousefi, SA, Behroozifar, M, and Dehghan, M, 2012. Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math Model. 36 (2012), pp. 945–963.
   Web of Science ®Google Scholar
• Yousefi, SA, Barikbin, Z, and Behroozifar, M, 2010. Bernstein Ritz–Galerkin method for solving the damped generalized regularized long-wave (DGRLW) equation, Int. J. Nonlinear Sci. 9 (2010), pp. 151–158.
   Google Scholar
• Yousefi, SA, Barikbin, Z, and Dehghan, M, 2010. Bernstein–Ritz–Galerkin method for solving an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Diff. Eqns. 26 (2010), pp. 1236–1246.
   Web of Science ®Google Scholar
• Stoer, J, and Bulirsch, R, 1980. Introduction to Numerical Analysis. New York: Springer-Verlag; 1980.
   Google Scholar
• Kirsch, A, 1999. An Introduction to the Mathematical Theory of Inverse Problem. New York: Springer; 1999.
   Google Scholar
• Krawczyk-Stando, D, and Rudnicki, M, 2007. Regularization parameter selection in discrete ill-posed problems-the use of the U-curve, Int. J. Appl. Math. Comput. Sci. 17 (2) (2007), pp. 157–164.
   Web of Science ®Google Scholar
• Johansson, BT, Lesnic, D, and Reeve, T, 2011. Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions, Inverse Probl. Sci. Eng. 19 (2011), pp. 659–677.
   Web of Science ®Google Scholar
• Hansen, PC, 1992. Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992), pp. 561–580.
   Web of Science ®Google Scholar
• Brent, RP, 2002. Algorithms for Minimization Without Derivatives. Mineola, NY: Dover; 2002.
   Google Scholar
• More, JJ, and Thuente, DJ, 1994. Line search algorithms with fauranteed sufficient decrease, ACM T. Math. Software. 20 (3) (1994), pp. 286–307.
   Web of Science ®Google Scholar
• More, JJ, and Weight, SJ, 1993. Optimization Software Guide. Philadelphia: SIAM; 1993.
   Google Scholar
• Dennis, JE, and Schnabel, RB, 1996. Numerical Methods for Unconstrained Optimization. Philadelphia: SIAM; 1996.
   Google Scholar
• Rheinboldt, WC, 1998. Methods for Solving Systems of Nonlinear Equations. Philadelphia: SIAM; 1998.
   Google Scholar
• Lakestani, M, and Dehghan, M, 2009. Numerical solution of Fokker–Plank equation using the cubic B-spline scaling functions, Numer. Methods Partial Diff. Eqns. 25 (2009), pp. 418–429.
   Web of Science ®Google Scholar
• Nevels, RD, Goswami, JC, and Tehrani, H, 1997. Semiorthogonal wavelets basis sets for solving integral equations, IEEE Trans. Antennas. Propagation 45 (1997), pp. 1332–1339.
   Web of Science ®Google Scholar
• Shamsi, M, and Dehghan, M, 2007. Recovering a time-dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method, Numer. Methods Partial Diff. Eqns. 23 (2007), pp. 196–210.
   Web of Science ®Google Scholar
• Elden, L, 1983. "The numerical solution of a non-characteristic Cauchy problem for a parabolic equation". In: Deuflhard, P, and Hairer, E, eds. Numerical Treatment of Inverse Problems in Differential and Integral Equations. Basel: Birkhauser; 1983. pp. 246–268.
   Google Scholar
• Johansson, BT, Lesnic, D, and Reeve, T, 2011. A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model. 35 (2011), pp. 4367–4378.
   Web of Science ®Google Scholar
• Adams, L, and Nazareth, JL, 1996. Linear and Nonlinear Conjugate Gradient-related Methods. Philadelphia: SIAM; 1996.
   Google Scholar