AM  Vol.2 No.11 , November 2011
An Introduction to Numerical Methods for the Solutions of Partial Differential Equations
ABSTRACT
Partial differential equations arise in formulations of problems involving functions of several variables such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity, etc. The present paper deals with a general introduction and classification of partial differential equations and the numerical methods available in the literature for the solution of partial differential equations.

Cite this paper
nullM. Kumar and G. Mishra, "An Introduction to Numerical Methods for the Solutions of Partial Differential Equations," Applied Mathematics, Vol. 2 No. 11, 2011, pp. 1327-1338. doi: 10.4236/am.2011.211186.
References
[1]   M. Kline, “Mathematical Thought from Ancient to Modern Times,” Oxford University Press, London, 1972.

[2]   H. Poincare, “Sur les Equations aux Derivees Partielles de la Physique Mathematique,” American Journal of Mathematics, Vol. 12, No. 3, 1890, pp. 211-294. doi:10.2307/2369620

[3]   H. Poincare, “Sur les Equations de la Physique Mathe-Matique,” Rendiconti del Circolo Mathematico di Palermo, Vol. 8, 1894, pp. 57-155.

[4]   E. E. Levi, “Sulle Equazioni Lineare Totalmente Ellitiche,” Rendiconti del Circolo Mathematico di Palermo, Vol. 24, No. 1, 1907, pp. 275-317. doi:10.1007/BF03015067

[5]   O. A. Ladyzhenskaya and N. N. Ural’tseva, “Linear and Quasi-Linear Elliptic Equations,” Academic Press, New York, 1968.

[6]   A. M. Micheletti and A. Pistoia, “On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds,” International Journal of Differential Equations, Vol. 2010, 2010, pp. 1-11. doi:10.1155/2010/432759

[7]   I. M. Gelfand, “Some Problems in Theory of Quasilinear Equations,” Transactions of the American Mathematical Society, Vol. 29, 1963, pp. 295-381.

[8]   D. D. Joseph and E. M. Sparrow, “Nonlinear Diffusion Induced by Nonlinear Sources,” Quarterly of Applied Mathematics, Vol. 28, 1970, pp. 327-342.

[9]   H. B. Keller and D.S. cohen, “Some Positive Problems Suggested by Nonlinear Heat Generation,” Journal of Mathematics and Mechanics, Vol. 16, No. 12, 1967, pp. 1361-1376.

[10]   G. E. Forsthye and W. R. Wasow, “Finite Difference Methods for Partial Differential Equations,” Wiley, New York, 1960.

[11]   J. D. Hoffman, “Numerical Methods for Engineers and Scientists,” 2nd Edition, McGraw-Hill, Inc., New York, 1992.

[12]   M. K. Jain, R. K. Jain, “R. K. Mohanty, Fourth Order Difference Methods for the System of 2-D Non-Linear Elliptic Partial Differential Equations,” Numerical Methods for Partial Differential Equations, Vol. 7, No. 3, 1991, pp. 227-244. doi:10.1002/num.1690070303

[13]   L. V. Kantorovich and V. I. Krylov, “Approximate Methods in Higher Analysis,” 3rd Edition, Interscience, New York, 1958.

[14]   M. Kumar, P. Singh and P. Kumar, “A Survey on Various Computational Techniques for Nonlinear Elliptic Boundary Value Problems,” Advances in Engineering Software, Vol. 39, No. 9, 2008, pp. 725-736. doi:10.1016/j.advengsoft.2007.11.001

[15]   M. Kumar and P. Kumar, “Computational Method for Finding Various Solutions for a Quasilinear Elliptic Equation of Kirchhoff Type,” Advances in Engineering Software, Vol. 40, No. 11, 2009, pp. 1104-1111. doi:10.1016/j.advengsoft.2009.06.003

[16]   R. K. Mohanty and S. Dey, “A New Finite Difference Discretization of Order Four for for Two-Dimensional Quasi-Linear Elliptic Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 76, No. 4, 2001, pp. 505-516. doi:10.1080/00207160108805043

[17]   L. A. Ogenesjan and L. A. Ruchovec, “Study of the Rate of Convergence of Variational Difference Schemes for Second-Order Elliptic Equations in a Two-Dimensional Field with a Smooth Boundary,” USSR Computational Mathematics and Mathematical Physics, Vol. 9, No. 5, 1969, pp. 158-183. doi:10.1016/0041-5553(69)90159-1

[18]   R. D. Richtmyer and K. W. Morton, “Difference Methods for Initial Value Problems,” 2nd Edition, Wiley-Interscience, New York, 1967.

[19]   A. A. Samarskii, “Theory of Difference Schemes,” Marcel Dekker Inc., New York, 2001.

[20]   R. Eymard and T. R. Gallou?t and R. Herbin, “The Finite Volume Method Handbook of Numerical Analysis,” Vol. 7, 2000, pp. 713-1020. doi:10.1016/S1570-8659(00)07005-8

[21]   R. J. Leveque, “Finite Volume Methods for Hyperbolic Problems,” Cambridge University Press, Cambridge, 2002.

[22]   P. Wesseling, “Principles of Computational Fluid Dynamics,” Springer-Verlag, Berlin, 2001. doi:10.1007/978-3-642-05146-3

[23]   I. Babuska, “Courant Element: Before and After,” In: M. Krizek, P. Neittanmaki and R. Stenberg, Eds., Finite Element Methods: Fifty Years of the Courant Element, Marcel Dekker, New York, 1994, pp. 37-57.

[24]   A. Pedasa and E. Tamme, “Discrete Galerkin Method for Fredholm Integro-Differential Equations with Weakly Singular Kernel,” Journal of Computational and Applied Mathematics, Vol. 213, No. 1, 2008, pp. 111-126. doi:10.1016/j.cam.2006.12.024

[25]   R. P. Kulkarni and N. Gnaneshwar, “Iterated Discrete Polynomially Based Galerkin Methods,” Applied Mathematics and Computation, Vol. 146, No. 1, 2003, pp. 153-165. doi:10.1016/S0096-3003(02)00533-7

[26]   M. H. Schultz, “Rayleigh-Ritz-Galerkin Methods for Multi-Dimensional Problems,” SIAM Journal on Numerical Analysis, Vol. 6, No. 4, 1969, pp. 523-538. doi:10.1137/0706047

[27]   M. H. Schultz, “L2 Error Bounds for the Rayleigh-Ritz-Galerkin Method,” SIAM Journal on Numerical Analysis, Vol. 8, No. 4, 1971, pp. 737-748. doi:10.1137/0708067

[28]   Y. Jianga and Y. Xu, “Fast Fourier Galerkin Methods for Solving Singular Boundary Integral Equations: Numerical Integration and Precondition,” Journal of Computational and Applied Mathematics, Vol. 234, No. 9, 2010, pp. 2792-2807. doi:10.1016/j.cam.2010.01.022

[29]   I. Babuska and A. K. Aziz, “Survey Lectures on the Mathematical Foundation of the Finite Element Method,” In: A. K. Aziz, Ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 5-359.

[30]   K. B?hmer, “Numerical Methods for Nonlinear Elliptic Differential Equations,” Oxford University Press, New York, 2010.

[31]   R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and Vibration,” Bulletin of American, Mathematical Society, Vol. 49, 1943, pp. 1-23. doi:10.1090/S0002-9904-1943-07818-4

[32]   M. Ghimenti and A. M. Micheletti, “On the Number of Nodal Solutions for a Nonlinear Elliptic Problems on Symmetric Riemannian Manifolds,” Electronic Journal of Differential Equations, Vol. 18, 2010, pp. 15-22.

[33]   N. Hirano, “Multiple Existence of Solutions for a Nonlinear Elliptic Problem on a Riemannian Manifold,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 70, No. 2, 2009, pp. 671-692.

[34]   D. V. Hutton, “Fundamentals of Finite Element Analysis,” Tata McGraw-Hill, New York, 2005.

[35]   M. Kumar and P. Kumar, “A Finite Element Approach for Finding Positive Solutions of Semilinear Elliptic Dirichlet Problems,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 5, 2009, pp. 1119-1128. doi:10.1002/num.20390

[36]   M. Kumar and P. Kumar, “Simulation of a Nonlinear Steklov Eigen Value Problem Using Finite Element Approximation,” Computational Mathematics and Modelling, Vol. 21, No.1, 2010, pp. 109-116. doi:10.1007/s10598-010-9058-6

[37]   R. Molle, “Semilinear Elliptic Problems in Unbounded Domains with Unbounded Boundary,” Asymptotic Analysis, Vol. 38, No. 3-4, 2004, pp. 293-307.

[38]   J. T. Oden and D. Somogyi, “Finite Element Applications in Fluid Dynamics,” Journal of the Engineering Mechanics Division, ASCE, Vol. 95, No. 4, 1968, pp. 821-826.

[39]   J. T. Oden, “A General Theory of Finite Elements, II: Applications,” International Journal for Numerical Meth- ods in Engineering, Vol. 1, No. 3, 1969, pp. 247-259. doi:10.1002/nme.1620010304

[40]   J. T. Oden, “A Finite Element Analogue of the Navier-Stokes Equations,” Journal of the Engineering Mechanics Division, ASCE, Vol. 96, No. 4, 1970, pp. 529-534.

[41]   E. R. De Arantes and E. Oliveira, “Theoretical Foundation of the Finite Element Method,” International Journal of Solids and Structures Vol. 4, No. 10, 1968, pp. 926-952.

[42]   S. S. Rao, “The Finite Element Method in Engineering,” 4th Edition, Elsevier, Butterworth Heinemann, 2005.

[43]   J. N. Reddy, “An Introduction to the Finite Element Method,” 3rd Edition, McGraw-Hill, New York, 2005.

[44]   M. Ramos and H. Tavares, “Solutions with Multiple Spike Patterns for an Elliptic System,” Calculus of Variations and Partial Differential Equations, Vol. 31, No. 1, 2008, pp. 1-25. doi:10.1007/s00526-007-0103-z

[45]   F. Williamson, “A Historical Note on the Finite Element Method,” International Journal for Numerical Methods in Engineering, Vol. 15, No. 6, 1980, pp. 930-934. doi:10.1002/nme.1620150611

[46]   O. C. Zienkiewicz, “The Finite Element Method in Engineering Science,” 3rd Edition, Mc-Graw-Hill, London, 1977.

[47]   M. Zlamal, “On the Finite Element Method,” Numerische Mathematik, Vol. 12, No. 5, 1968, pp. 394-409. doi:10.1007/BF02161362

 
 
Top