Numerical Methods Based on Non-Polynomial Sextic Spline for Solution of Variable Coefficient Fourth-Order Wave Equations

REFERENCES

• Meirovitch , L. 1997 . Principles and Techniques of Vibrations , Englewood Cliffs, NJ : Prentice Hall, Inc. .
   Google Scholar
• Meirovitch , L. 1967 . Analytical Methods in Vibrations , New York : Macmillan .
   Google Scholar
• Thomson , W. T. 1988 . Theory of Vibration with Applications, , 3rd ed. , Englewood : Prentice Hall .
   Google Scholar
• Gorman , D. J. 1975 . Free Vibrations Analysis of Beams and Shafts , New York : John Wiley & Sons .
   Google Scholar
• Jain , M. K. 1984 . Numerical Solution of Differential Equations, , 2 ed. , New Delhi, , India : Wiley Eastern .
   Google Scholar
• Richtmyer , R. D. and Morton , K. W. 1967 . Difference Methods for Initial Value Problems, , 2nd ed. , New York : Wiley-Interscience .
   Google Scholar
• Fairweather , G. and Gourlay , A. R. 1967 . Some Stable Difference Approximations to a Fourth Order Parabolic Partial Differential Equation . Math. Comput. , vol. 21 : 1 – 11 .
   Web of Science ®Google Scholar
• Danaee , A. and Evans , D. J. 1982 . Hopscotch Procedures for a Fourth-Order Parabolic Partial Differential Equation . Math. Computers Simul. , XXIV : 326 – 329 .
   Google Scholar
• Evans , D. J. 1965 . A Stable Explicit Method for the Finite Difference Solution of a Fourth Order Parabolic Partial Differential Equation . Comput. J. , vol. 8 : 280 – 287 .
   Web of Science ®Google Scholar
• Collatz , L. 1973 . “ Hermitian Methods for Initial Value Problems in Partial Differential Equations ” . In Topics in Numerical Analysis , Edited by: Miller , J. J. H. 41 – 61 . New York : Academic Press .
   Google Scholar
• Todd , J. 1956 . A Direct Approach to the Problem of Stability in the Numerical Solution of Partial Differential Equations . Commun. Pure Appl. Math. , vol. 9 : 597 – 612 .
   Web of Science ®Google Scholar
• Crandall , S. H. 1954 . Numerical Treatment of a Fourth Order Partial Differential Equations . Assoc. J. Comput. Mech. , vol. 1 : 111 – 118 .
   Google Scholar
• Albrecht , J. 1957 . Zum Differenzenverfahren bei Parabolischen Differentialgleichungen . Z. Angew. Math. Mech. , vol. 37 : 202 – 212 .
   Google Scholar
• Jain , M. K. , Iyengar , S. R. K. and Lone , A. G. 1976 . Higher Order Difference Formulas for a Fourth Order Parabolic Partial Differential Equation . Intern. J. Numer. Methods Eng. , vol. 10 : 1357 – 1367 .
   Web of Science ®Google Scholar
• Rashidinia , J. 1994 . Applications of Spline to Numerical Solution of Differential Equations , Ph. D thesis India : Aligarh Muslim University .
   Google Scholar
• Rashidinia , J. and Aziz , T. 1998 . Spline Solution of Fourth-Order Parabolic Partial Differential Equations . Intern. J. Appl. Sci. Comput. , vol. 5 : 139 – 148 .
   Google Scholar
• Andrade , C. and McKee , S. 1977 . High Accuracy A. D. I. Methods for Fourth Order Parabolic Equations with Variable Coefficients . J. Comput. Appl. Math. , vol. 3 : 11 – 14 .
   Google Scholar
• Evans , D. J. and Yousif , W. S. 1991 . A Note on Solving the Fourth Order Parabolic Equation by the Age Method . Intern. J. Computer Math. , vol. 40 : 93 – 97 .
   Web of Science ®Google Scholar
• Khaliq , A. Q. M. and Twizell , E. H. 1987 . A Family of Second Order Methods for Variable Coefficient Fourth Order Parabolic Partial Differential Equations . Intern. J. Computer Math. , vol. 23 : 63 – 76 .
   Web of Science ®Google Scholar
• Wazwaz , Abdul-Majid . 2001 . Analytic Treatment for Variable Coefficient Fourth-Order Parabolic Partial Differential Equations . Appl. Math. Comput. , vol. 123 : 219 – 227 .
   Web of Science ®Google Scholar
• Aziz , T. , Khan , A. and Rashidinia , J. 2005 . Spline Methods for the Solution of Fourth-Order Parabolic Partial Differential Equations . Appl. Math. Comput. , vol. 167 : 153 – 166 .
   Web of Science ®Google Scholar
• Khan , A. , Khan , I. and Aziz , T. 2005 . Sextic Spline Solution for Solving a Fourth-Order Parabolic Partial Differential Equation . Intern. J. Computer Math. , vol. 82 : 871 – 879 .
   Web of Science ®Google Scholar
• Rashidinia , J. , Mohammadi , R. and Jalilian , R. 2007 . Spline Methods for the Solution of Hyperbolic Equation with Variable Coefficients . Numer. Methods Partial Differential Eqs. , vol. 23 : 1411 – 1419 .
   Web of Science ®Google Scholar