**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30761

##### Quintic Spline Solution of Fourth-Order Parabolic Equations Arising in Beam Theory

**Authors:**
Reza Mohammadi,
Mahdieh Sahebi

**Abstract:**

**Keywords:**
Stability Analysis,
fourth-order parabolic equation,
variable coefficient,
polynomial quintic spline,
off-step points

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1125773

**References:**

[1] A. Q. M. Khaliq and E. H. Twizell, A family of second order methods for variable coefficient fourth order parabolic partial differential equations, Intern. J. Computer Math. 23 (1987) 63-76.

[2] D. J. Gorman, Free Vibrations Analysis of Beams and Shafts, John Wiley & Sons, New York, 1975.

[3] M. K. Jain, S. R. K. Iyengar and A. G. Lone, Higher order difference formulas for a fourth order parabolic partial differential equation, Intern. J. Numer. Methods Eng. 10 (1976) 1357-1367.

[4] R. D. Richtmyer and K. W. Mortan, Difference Methods for Initial Value Problems, (2nd ed.) (NewYork: Wiley-Interscience), (1967).

[5] G. Fairweather and A. R. Gourlay, Some stable difference approximations to a fourth order parabolic partial differential equation, Math. Comput. 21 (1967) 1-11.

[6] A. Danaee and D. J. Evans, Hopscotch procedures for a fourth-order parabolic partial differential equation, Math. Computers Simul. XXIV (1982) 326-329.

[7] D. J. Evans, A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation, Comput. J. 8 (1965) 280-287.

[8] L. Collatz, Hermitian methods for initial value problems in partial differential equations, In: J.J.H. Miller (Ed.) Topics in Numerical Analysis (NewYork: Academic Press), (1973) 41-61.

[9] C. Andrade and S. McKee, High accuracy A.D.I. methods for fourth order parabolic equations with variable coefficients, J. Comput. Appl. Math. 3 (1) (1977) 11-14.

[10] D. J. Evans and W. S. Yousif, A note on solving the fourth order parabolic equation by the age method, Intern. J. Computer Math. 40 (1991) 93-97.

[11] J. Albrecht, Zum Differenzenverfahren bei parabolischen Differentialgleichungen, Z. Angew. Math. Mech., 37 (1957) 202-212.

[12] S. H. Crandall, Numerical treatment of a fourth order partial differential equations, J. Assoc. Comput. Mech. 1 (1954) 111-118.

[13] M. K. Jain, Numerical Solution of Differential Equations, Second Ed., Wiley Eastern, New Delhi, India, 1984 .

[14] J. Todd, A direct approach to the problem of stability in the numerical solution of partial differential equations, Commun. Pure Appl. Math. 9 (1956) 597-612.

[15] J. Rashidinia, Applications of spline to numerical solution of differential equations, Ph. D Thesis, Aligarh Muslim University, India, 1994.

[16] J. Rashidinia and T. Aziz, Spline solution of fourth-order parabolic partial differential equations, Intern. J. Appl. Sci. Comput. 5 (2) (1998) 139-148.

[17] T. Aziz, A. Khan and J. Rashidinia, Spline methods for the solution of fourth-order parabolic partial differential equations, Appl. Math. Comput. 167 (2005) 153-166.

[18] A. Khan, I. Khan and T. Aziz, Sextic spline solution for solving a fourth-order parabolic partial differential equation, Intern. J. Computer Math. 82 (7) (2005) 871-879.

[19] Abdul-Majid Wazwaz, Analytic treatment for variable coefficient fourth-order parabolic partial differential equations, Appl. Math. Comput. 123 (2001) 219-227.

[20] J. Rashidinia, R. Mohammadi and R. Jalilian, Spline methods for the solution of hyperbolic equation with variable coefficients, Numer. Methods Partial Differential Eq. 23 (2007) 1411-1419.