Polynomial solution of singular differential equations using Weighted Sobolev gradients

References

• B. Brown, M. Jais, and I. Knowles, A variational approach to an elastic inverse problem, Inverse Probl. 21 (2005), pp. 1953–1973. doi: 10.1088/0266-5611/21/6/010
   Web of Science ®Google Scholar
• J.G. Dix and T.W. McCabe, On finding equilibria for isotropic hyperelastic materials, Nonlinear Anal 15 (1990), pp. 437–444. doi: 10.1016/0362-546X(90)90125-Z
   Web of Science ®Google Scholar
• J. Garcia-Ripoll, V. Konotop, B. Malomed, and V. Perez-Garcia, A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates, Math. Comput. Simul. 62 (2003), pp. 21–30. doi: 10.1016/S0378-4754(02)00190-8
   Web of Science ®Google Scholar
• J. Karatson and I. Farago, Preconditioning operators and Sobolev gradients for nonlinear elliptic problems, Comput. Math. Appl. 50 (2005), pp. 1077–1092. doi: 10.1016/j.camwa.2005.08.010
   Web of Science ®Google Scholar
• J. Karatson and L. Loczi, Sobolev gradient preconditioning for the electrostatic potential equation, Comput. Math. Appl. 50 (2005), pp. 1093–1104. doi: 10.1016/j.camwa.2005.08.011
   Web of Science ®Google Scholar
• I. Knowles and A. Yan, On the recovery of transport parameters in groundwater modelling, J. Comput. Appl. Math. 171 (2004), pp. 277–290. doi: 10.1016/j.cam.2004.01.038
   Web of Science ®Google Scholar
• A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, New York, 1985.
   Google Scholar
• D.G. Luenberger, Linear and Nonlinear Programming, Addison Wesley, Reading Massachusetts, 1989.
   Google Scholar
• W.T. Mahavier, Solving boundary value problems numerically using steepest descent in Sobolev spaces, Missouri J. Math. Sci. 11(1) (1999), pp. 19–32.
   Google Scholar
• W.T. Mahavier, A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonl. World 4 (1997).
   Google Scholar
• J.W. Neuberger, Projection methods for linear and nonlinear systems of partial differential equations, in Ordinary Differential Equations and Operators, Lecture Notes in Math. Vol. 546, Springer-Verlag, New York, 1976, pp. 341–349.
   Google Scholar
• J.W. Neuberger, Steepest descent for general systems of linear differential equations in Hilbert space, in Ordinary Differential Equations and Operators, Lecture Notes in Math. Vol. 1032, Springer-Verlag, New York, 1982, pp. 390–406.
   Google Scholar
• J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics 1670, Springer-Verlag, New York, 1997.
   Google Scholar
• J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
   Google Scholar
• A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, McGraw-Hill, New York, 1978.
   Google Scholar
• N. Raza, S. Sial, S. Siddiqi, and T. Lookman, Energy-minimization related to Nonlinear Schrödinger equation, J. Comput. Phys. 228 (2009), pp. 2572–2577. doi: 10.1016/j.jcp.2008.12.016
   Web of Science ®Google Scholar
• N. Raza, S. Sial, and S. Siddiqi, Simulation study of propagation of pulses in optical fibre communication systems using Sobolev gradients and split-step Fourier methods, Int. J. Comput. Methods 6 (2009), pp. 1–12. doi: 10.1142/S0219876209001772
   Web of Science ®Google Scholar
• N. Raza, S. Sial, and S. Siddiqi, Sobolev gradient approach for the time evolution related to energy minimization related to Ginzberg-Landau functionals, J. Comput. Phys. 228 (2009), pp. 2566–2571. doi: 10.1016/j.jcp.2008.12.017
   Web of Science ®Google Scholar
• N. Raza, S. Sial, and S. Siddiqi, Approximating time evolution related Ginzberg-Landau functionals via Sobolev gradient methods in a finite-element setting, J. Comput. Phys. 229 (2010), pp. 1621–1625. doi: 10.1016/j.jcp.2009.10.048
   Web of Science ®Google Scholar
• N. Raza, S. Sial, and J.W. Neuberger, Numerical solution of Burgers' equation by the Sobolev gradient method, Appl. Math. Comput. 218 (2011), pp. 4017–4024.
   Web of Science ®Google Scholar
• N. Raza, S. Sial, J.W. Neuberger, and M.O. Ahmad, Numerical solutions of integro-differential equations using Sobolev gradient method, Int. J. Comput. Methods 9 (2012), pp. 1250046. doi: 10.1142/S0219876212500466
   Web of Science ®Google Scholar
• N. Raza and S. Sial, Sobolev gradient approach for Huxley and Fisher models for gene propagation, Appl. Math. (Irvine) 4 (2013), pp. 1212–1219. doi: 10.4236/am.2013.48163
   Google Scholar
• R.J. Renka and J.W. Neuberger, Minimal surfaces and Sobolev gradients, SIAM J. Sci. Comput. 16 (1995), pp. 1412–1427. doi: 10.1137/0916082
   Web of Science ®Google Scholar
• S. Sial, J. Neuberger, T. Lookman, and A. Saxena, Energy minimization using Sobolev gradients: Application to phase separation and ordering, J. Comput. Phys. 189 (2003), pp. 88–97. doi: 10.1016/S0021-9991(03)00202-X
   Web of Science ®Google Scholar