The Generalized Solutions of Second and Third-Order Cauchy-Euler Eequations by Using the Elzaki Transforms
Abstract
This paper aims to study the generalized solutions of Cauchy-Euler equations of the form and where and are integers and using Elzaki transform technique. The solutions are in the space of distributions. Types of solutions are in the form of a distributional solution and a weak solution which depends on the values of and Keywords : Cauchy-Euler equation; Dirac delta function; Elzaki transform; The generalized solutions.References
Abbasbandy, S., & Eslaminasab, M. (2015). Study on usage of Elzaki transform for the ordinary differential
equations with non-constant coefficients. Journal of Industrial Mathematics, 7(3), 277-281.
Alderremy, A. (2018). A comments on the paper: New transform iterative method for solving some Klein Gordon
equations. Results in Physics, 11, 510-511.
Devi, A., Gill, V., & Roy, P. (2017). Solution of ordinary differential equations with variable coefficients using Elzaki
transform. Asian Journal of Applied Science and Technology, 1(9), 186-194.
Elzaki, T. M. & Elzaki, S. M. (2011). On the connections between Laplace and Elzaki transforms. Advances in
Theoretical and Applied Mathematics, 6(1), 1-11.
Elzaki, T. M. & Elzaki, S. M. (2011). On the Elzaki transform and ordinary differential equation with variable
coefficients. Advances in Theoretical and Applied Mathematics, 6(1), 41-43.
Elzaki, T. M. & Elzaki, S. M. (2012). On the new integral transform Elzaki transform fundamental properties
investigation and applications. Global Journal of Mathematical Sciences, 4(1), 1-13.
Elzaki, T. M., Elzaki, S. M., & Hilal, E. M. A. (2012). Elzaki and Sumudu transforms for solving some differential
equations. Global Journal of Pure & Applied Mathematics, 8(2), 167-173.
Ghil, B. & Kim, H. (2015). The solution of Euler-Cauchy equation using Laplace transform. International Journal of
Mathematical Analysis, 9(53), 2611 - 2618.
Jhanthanam, S., Kim, H., & Nonlaopon, K. (2019). Generalized solutions of the third-Order Cauchy-Euler equation
in the space of right-sided distributions via Laplace transform. Mathematics, 7(4), 376.
Kananthai, A. (1999). Distribution solutions of the third-order Euler equation. Southeast Asian Bulletin of Mathematics, 23, 627-631.
Kanwal, R.P. (2004). Generalized function, Theory and applications. (3rd ed). Massachusetts: Birkhauser Boston.
Kim, H. (2013). The time shifting theorem and the convolution for Elzaki transform. Global Journal of Pure and
Applied Mathematics, 87(2), 261-271.
Kim, H. (2014). The shifted data problems by using transform of derivatives. Applied Mathematical Sciences, 8(151), 149-152.
Kim, H. (2016). The method to find a basis of Euler-Cauchy equation by transforms. Global Journal of Pure and
Applied Mathematics, 12(5), 4159-4165.
Kim, H., Nonlaopon, K., & Sacorn, N. (2018). A note on the generalized solutions of the third-order Cauchy-Euler
equations. Communications in Mathematics and Applications, 9(4), 661-669.
Kim, H., & Song, Y. (2014). The solution of Voterra integral equation of the second kind by using the Elzaki
transform. Applied Mathematical Science, 8(11), 525-530.
equations with non-constant coefficients. Journal of Industrial Mathematics, 7(3), 277-281.
Alderremy, A. (2018). A comments on the paper: New transform iterative method for solving some Klein Gordon
equations. Results in Physics, 11, 510-511.
Devi, A., Gill, V., & Roy, P. (2017). Solution of ordinary differential equations with variable coefficients using Elzaki
transform. Asian Journal of Applied Science and Technology, 1(9), 186-194.
Elzaki, T. M. & Elzaki, S. M. (2011). On the connections between Laplace and Elzaki transforms. Advances in
Theoretical and Applied Mathematics, 6(1), 1-11.
Elzaki, T. M. & Elzaki, S. M. (2011). On the Elzaki transform and ordinary differential equation with variable
coefficients. Advances in Theoretical and Applied Mathematics, 6(1), 41-43.
Elzaki, T. M. & Elzaki, S. M. (2012). On the new integral transform Elzaki transform fundamental properties
investigation and applications. Global Journal of Mathematical Sciences, 4(1), 1-13.
Elzaki, T. M., Elzaki, S. M., & Hilal, E. M. A. (2012). Elzaki and Sumudu transforms for solving some differential
equations. Global Journal of Pure & Applied Mathematics, 8(2), 167-173.
Ghil, B. & Kim, H. (2015). The solution of Euler-Cauchy equation using Laplace transform. International Journal of
Mathematical Analysis, 9(53), 2611 - 2618.
Jhanthanam, S., Kim, H., & Nonlaopon, K. (2019). Generalized solutions of the third-Order Cauchy-Euler equation
in the space of right-sided distributions via Laplace transform. Mathematics, 7(4), 376.
Kananthai, A. (1999). Distribution solutions of the third-order Euler equation. Southeast Asian Bulletin of Mathematics, 23, 627-631.
Kanwal, R.P. (2004). Generalized function, Theory and applications. (3rd ed). Massachusetts: Birkhauser Boston.
Kim, H. (2013). The time shifting theorem and the convolution for Elzaki transform. Global Journal of Pure and
Applied Mathematics, 87(2), 261-271.
Kim, H. (2014). The shifted data problems by using transform of derivatives. Applied Mathematical Sciences, 8(151), 149-152.
Kim, H. (2016). The method to find a basis of Euler-Cauchy equation by transforms. Global Journal of Pure and
Applied Mathematics, 12(5), 4159-4165.
Kim, H., Nonlaopon, K., & Sacorn, N. (2018). A note on the generalized solutions of the third-order Cauchy-Euler
equations. Communications in Mathematics and Applications, 9(4), 661-669.
Kim, H., & Song, Y. (2014). The solution of Voterra integral equation of the second kind by using the Elzaki
transform. Applied Mathematical Science, 8(11), 525-530.
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Published
2023-01-04
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