A new method for exact product form and approximation solutions of a parabolic equation with nonlocal initial condition using Ritz method

Document Type : Research Article

Author

• Z. Barikbin

Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran,

10.22067/ijnao.v10i1.83685

Abstract

Many phenomena in various fields of physics are simulated by parabolic partial differential equations with the nonlocal initial conditions, while there are few numerical methods for solving these problems. In this paper, the Ritz–Galerkin method with a new approach is proposed to give the exact and approximate product solution of a parabolic equation with the nonstandard initial conditions. For this purpose, at first, we introduce a function called satisfier function, which satisfies all the initial and boundary conditions. The uniqueness of the satisfier function and its relation to the exact solution are discussed. Then the Ritz–Galerkin method with satisfier function is used to simplify the parabolic partial differential equations to the solution of algebraic equations. Error analysis is worked by using the property of interpolation. The comparisons of the obtained results with the results of other methods show more accuracy in the presented technique.

Keywords

• Nonlocal time weighting initial condition
• Ritz–Galerkin method
• Satisfier function
• Bernstein polynomials
• Numerical Solution
• error analysis

20.1001.1.24236977.2020.10.1.8.8

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References

[1] Barikbin, Z. Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion wave equation with damping, Math. Sci. 11 (2017), 195–202.

[2] Barikbin, Z., Ellahi, R. and Abbasbandy, S.The Ritz–Galerkin method for MHD Couette flow of non-Newtonian fluid, Int. J. Ind. Math. 6 (2014), 235–243.

[3] Bysezewski, L. Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal. 40 (1991), 173–180.

[4] Bysezewski, L. Uniqueness of solutions of parabolic semilinear nonlinear boundary problems, J. Math. Anal. Appl. 165 (1992), 427–478.

[5] Bysezewski, L. and Lakshmikantham, V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1990), 11–19.

[6] Cahlon, B., Kulkarni, D.M. and Shi, P. Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. Numer. Anal. 32 (1995), 571–593.

[7] Cannon, J.R. The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155–160.

[8] Cannon, J.R. The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley, Publishing Com pany, Menlo Park, CA, (1984).

[9] Chabrowski, J. On non-local problems for parabolic equations, Nagoya Math. J. 93 (1984), 109–131.

[10] Chadam, J.M. and Yin, H.-M. Determination of an unknown function in a parabolic equation with an overspecified condition, Math. Methods Appl. Sci. 13 (1990), 421–430.

[11] Čiupaila, R., Sapagovas, M. and Štikonienė, O. Numerical solution of nonlinear elliptic equation with nonlocal condition, Non- linear Anal. Model. Control. 18 (4) (2013), 412–426.

[12] Choi, Y.S. and Chan, K.Y. A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 18 (1992), 317– 331.

[13] Dehghan, M. Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition, Appl. Math. Comput. 147 (2)(2004), 321–331.

[14] Dehghan, M. Three-level techniques for one-dimensional parabolic equation with nonlinear initial condition, Appl. Math. Comput. 151 (2)(2004), 567–579.

[15] Dehghan, M. Implicit collocation technique for heat equation with non classic initial condition, Int. J. Non-Linear Sci. Numer. Simul. 7 (2006), 447–450.

[16] Diaz, J. I., Padial, J.F. and Rakotoson, J.M. Mathematical treatment of the magnetic confinement in a current carrying stellarator, Nonlinear Anal. 34 (6) (1998), 857–887.

[17] Ewing, R.E., Lazarov, R.D. and Lin, Y. Finite volume element approximations of nonlocal in time one-dimensional flows in porous media, Computing, 64 (2) (2000), 157–182.

[18] Gasca, M. and Sauer, T. On the history of multivariate polynomial in terpolation, J. Comput. Appl. Math. 122 (2000), 23–35.

[19] Glotov, D.W., Hames, E., Meirc, A.J. and Ngoma, S. An integral constrained parabolic problem with applications in thermochronology, Comp. Math. Appl. 71 (11) (2016), 2301–2312.

[20] Kamynin, N.I. A boundary value in the theory of the heat conduction with non-local boundary condition, USSR Comput. Math. Math. Phys. 4 (1964), 33–59.

[21] Lesnic, D., Yousefi, S.A. and Ivanchov, M. Determination of a time dependent diffusivity from nonlocal conditions, J. Appl. Math. Comput. 41 (2013), 301–320.

[22] Martin-Vaquero, J. and Sajavičius, S. The two-level finite difference schemes for the heat equation with nonlocal initial condition, Appl. Math. Comput. 342 (2019), 166–177.

[23] Mason, J.C. and Handscomb, D.C. Chebyshev polynomials, CRC Press LLC, 2003.

[24] Olmstead, W.E. and Roberts, C.A. The one-dimensional heat equation with a nonlocal initial condition, Appl. Math. Lett. 10 (3) (1997), 89–94.

[25] Pao, C.V. Reaction diffusion equations with nonlocal boundary and non local initial conditions, J. Math. Anal. Appl. 195 (3) (1995), 702–718.

[26] Rashedi, K., Adibi, H. and Dehghan, M. Determination of space-time dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique, Inverse Probl. Sci. Eng. 22 (2014), 1077–1108.

[27] Schultz, M.H. Error bounds for the Rayleigh-Ritz–Galerkin method, J. Math. Anal. Appl. 27 (1969), 524–533.

[28] Schultz, M.H. L2 Error Bounds for the Rayleigh-Ritz- Galerkin Method, Siam. J. Numer. Anal. 8 (1971), 737–748.

[29] Shelukhin, V.V. A non-local (in time) model for radionuclides propagation in a Stokes fluid, Dinamika Sploshn, Sredy, 107 (1993), 180–193.

[30] Shi, P. Weak solution to evolution problem with a nonlocal constraint, SIAM J. Math. Anal. 24 (1993), 46–58.

[31] Shimin, G., Liquan, M., Zhengqiang, Z. and Yutao, J. Finite difference/spectral-Galerkin method for a two-dimensional distributed order time-space fractional reaction-diffusion equation, Appl. Math. Lett. 85 (2018), 157–163.

[32] Vinogradova, P. and Zarubin, A. A study of Galerkin method for the heat convection equations, Appl Math. 57 (2012), 71–91.

[33] Yousefi, S.A. and Barikbin, Z. Ritz Legendre Multiwavelet method for the Damped Generalized Regularized Long-Wave equation, J. Comput. Nonlinear Dyn. 7 (2011), 1–4.

[34] Yousefi, S.A., Barikbin, Z. and Dehghan, M. 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 (2012), 39–48.

[35] Yousefi, S.A., Lesnic, D. and Barikbin, Z. Satisfier function in Ritz–Galerkin method for the identification of a time-dependent diffusivity, J. Inverse Ill-Posed Probl. 20 (2012), 701–722.