References
- R. Almedia, R. Khosravian-Arab and M. Shamsi. A generalized fractional variational problem depending on indefinite integrals: Euler-Lagrange equation and numerical solution. J. Vib. Cont., 19(14):2177–2186, 2012. http://dx.doi.org/10.1177/1077546312458818.
- N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68:337–404, 1950. http://dx.doi.org/10.1090/S0002-9947-1950-0051437-7.
- O. Abu Arqub and M. Al-Smadi. Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations. Appl. Math. Comput., 243:911–922, 2014. http://dx.doi.org/10.1016/j.amc.2014.06.063.
- O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat. Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Computing., pp. 1–20, 2015. http://dx.doi.org/10.1007/s00500-015-1707-4.
- A. Berlinet and C.T. Agnan. Reproducing Kernel Hilbert Space in Probability and Statistics. Kluwer Academic Publishers, Boston, Mass, USA, 2004. http://dx.doi.org/10.1007/978-1-4419-9096-9.
- P. Bouboulis and M. Mavroforakis. Reproducing kernel Hilbert spaces and fractal interpolation. J. Comput. Appl. Math., 235(12):3425–3434, 2011. http://dx.doi.org/10.1016/j.cam.2011.02.003.
- M. Cui and F. Geng. A computational method for solving one-dimensional variable-coefficient Burgers equation. Appl. Math. Comput., 188(2):1389–1401, 2007. http://dx.doi.org/10.1016/j.amc.2006.11.005.
- M. Cui and F. Geng. Solving singular two-point boundary value problem in reproducing kernel space. J. Comput. Appl. Math., 205(1):6–15, 2007. http://dx.doi.org/10.1016/j.cam.2006.04.037.
- M. Cui and Y. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, New York, NY, USA, 2008.
- M. Cui and L. Yang. A new method of solving the coefficient inverse problem of differential equation. Science in China A., 50(4):561–572, 2007. http://dx.doi.org/10.1007/s11425-007-0013-8.
- A. Daniel. Reproducing Kernel Spaces and Applications. Springer, Basel, Switzerland, 2003.
- L.E. Elgolic. Calculus of Variations. Pergamon press, Oxford, 1962.
- I.M. Gelfand and S.V. Fomin. Calculus of Variations. Prentice-Hall, New Jersey, 1963.
- F. Geng. Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Appl. Math. Comput., 215(6):2095–2102, 2009. http://dx.doi.org/10.1016/j.amc.2009.08.002.
- F. Geng. A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method. Appl. Math. Comput., 218(8):4211–4215, 2011. http://dx.doi.org/10.1016/j.amc.2011.09.052.
- F. Geng and M. Cui. Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl., 327(2):1167–1181, 2007. http://dx.doi.org/10.1016/j.jmaa.2006.05.011.
- F. Geng and M. Cui. Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl. Math. Comput., 192(2):389–398, 2007. http://dx.doi.org/10.1016/j.amc.2007.03.016.
- F. Geng and M. Cui. A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett., 25(5):818–823, 2012. http://dx.doi.org/10.1016/j.aml.2011.10.025.
- F. Geng and X.M. Li. A new method for Riccati differential equations based on reproducing kernel and quasilinearization methods. Abs. Appl. Anal., p. 8 pages, 2012. http://dx.doi.org/10.1155/2012/603748.
- M. Ghasemi, M. Fardi and R. Khoshsiar Ghaziani. Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl. Math. Comput., 268:815–831, 2015. http://dx.doi.org/10.1016/j.amc.2015.06.012.
- I. Gyongy. Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process Appl., 73(2):271–299, 1998. http://dx.doi.org/10.1016/S0304-4149(97)00103-8.
- W. Jiang and Z. Chen. A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation. Numer. Meth. Part. Diff. Equ., 30(1):289–300, 2014. http://dx.doi.org/10.1002/num.21809.
- M.L. Krasnov, G.I. Makarenko and A.I. Kiselev. Problems and Exercises in the Calculus of Variations. MIR Publishers, Moscow, 1975.
- M. Mohammadi and R. Mokhtari. A reproducing kernel method for solving a class of nonlinear systems of PDEs. Math. Model. Anal., 19(2):180–198, 2014. http://dx.doi.org/10.3846/13926292.2014.909897.
- S. Momani, O. Abu Arqub, T. Hayat and H. Al-Sulami. A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Volterra type. Appl. Math. Comput., 240:229–239, 2014. http://dx.doi.org/10.1016/j.amc.2014.04.057.
- V.M. Tikhomirov. Store about Maxima and Minima, American Mathematica Society. Providence, RI, 1990.
- J. Wang and G. Warnecke. Existence and uniqueness of solutions for a non-uniformly parabolic equation. J. Diff. Equ., 183(1):1–16, 2003. http://dx.doi.org/10.1016/S0022-0396(02)00059-1.
- L. Yang and M. Cui. New algorithm for a class of nonlinear integro-differential equations in the reproducing kernel space. Appl. Math. Comput., 174(2):942–960, 2006. http://dx.doi.org/10.1016/j.amc.2005.05.026.
- Y. Yang, M. Du, F. Tan, Z. Li and T. Nie. Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions. Appl. Math. Comput., 219(11):5918–5925, 2013. http://dx.doi.org/10.1016/j.amc.2012.12.009.
- Y.F. Zho, M.G. Cui and Y.Z. Lin. A computational method for nonlinear 2mth order boundary value problems. Math. Model. Anal., 15(4):571–586, 2010. http://dx.doi.org/10.3846/1392-6292.2010.15.571-586.
- D.X. Zhou. Capacity of reproducing kernel spaces in learning theory. IEEE Transactions on Information Theory, 49(7):1743–1752, 2003. http://dx.doi.org/10.1109/TIT.2003.813564.