Hyers-Ulam-Rassias Stability for the Heat Equation

Author(s)
Maher Nazmi Qarawani

ABSTRACT

In this paper we apply the Fourier transform to prove the Hyers-Ulam-Rassias stability for one dimensional heat equation on an infinite rod. Further, the paper investigates the stability of heat equation in with initial condition, in the sense of Hyers-Ulam-Rassias. We have also used Laplace transform to establish the modified Hyers-Ulam-Rassias stability of initial-boundary value problem for heat equation on a finite rod. Some illustrative examples are given.

In this paper we apply the Fourier transform to prove the Hyers-Ulam-Rassias stability for one dimensional heat equation on an infinite rod. Further, the paper investigates the stability of heat equation in with initial condition, in the sense of Hyers-Ulam-Rassias. We have also used Laplace transform to establish the modified Hyers-Ulam-Rassias stability of initial-boundary value problem for heat equation on a finite rod. Some illustrative examples are given.

Cite this paper

M. Qarawani, "Hyers-Ulam-Rassias Stability for the Heat Equation,"*Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 1001-1008. doi: 10.4236/am.2013.47137.

M. Qarawani, "Hyers-Ulam-Rassias Stability for the Heat Equation,"

References

[1] D. H. Hyers, “On the Stability of the Linear Functional Equation,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 27, No. 4, 1941, pp. 222-224. doi:10.1073/pnas.27.4.222

[2] T. M. Rassias, “On the Stability of the Linear Mapping in Banach Spaces,” Proceedings of the American Mathematical Society, Vol. 72, No. 2, 1978, pp. 297-300. doi:10.1090/S0002-9939-1978-0507327-1

[3] P. Gavruta, “A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 184, No. 3, 1994, pp. 431-436. doi:10.1006/jmaa.1994.1211

[4] S. M. Jung, “On the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 1, 1996, pp. 221-226. doi:10.1006/jmaa.1996.0433

[5] S. M. Jung, “Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis,” Hadronic Press, Palm Harbor, 2001.

[6] T. Miura, S.-E. Takahasi and H. Choda, “On the HyersUlam Stability of Real Continuous Function Valued Differentiable Map,” Tokyo Journal of Mathematics, Vol. 24, No. 2, 2001, pp. 467-476. doi:10.3836/tjm/1255958187

[7] C. G. Park, “On the Stability of the Linear Mapping in Banach Modules,” Journal of Mathematical Analysis and Applications, Vol. 275, 2002, pp. 711-720. doi:10.1016/S0022-247X(02)00386-4

[8] C. G. Park, “Homomorphisms between Poisson JC*-Algebras,” Bulletin of the Brazilian Mathematical Society, Vol. 36, No. 1, 2005, pp. 79-97. doi:10.1007/s00574-005-0029-z

[9] C. G. Park, Y.-S. Cho and M. Han, “Functional Inequalities Associated with Jordan-Von Neumanntype Additive Functional Equations,” Journal of Inequalities and Applications, Vol. 2007, 2007, Article ID: 41820.

[10] K.-W. Jun and Y.-H. Lee, “A Generalization of the HyersUlam-Rassias Stability of the Pexiderized Quadratice Quations,” Journal of Mathematical Analysis and Applications, Vol. 297, No. 1, 2004, pp. 70-86. doi:10.1016/j.jmaa.2004.04.009

[11] C. Alsina and R. Ger, “On Some Inequalities and Stability Results Related to the Exponential Function,” Journal of Inequalities and Applications, Vol. 2, No. 4, 1998, pp. 373-380.

[12] E. Takahasi, T. Miura and S. Miyajima, “On the HyersUlam Stability of the Banach Space-Valued Differential Equation ,” Bulletin of the Korean Mathematical Society, Vol. 39, No. 2, 2002, pp. 309-315. doi:10.4134/BKMS.2002.39.2.309

[13] T. Miura, S. Miyajima and S.-E. Takahasi, “A Characterization of Hyers-Ulam Stability of First Order Linear Differential Operators,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 1, 2007, pp. 136-146. doi:10.1016/S0022-247X(03)00458-X

[14] S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 1, 2005, pp. 139146. doi:10.1016/j.jmaa.2005.02.025

[15] G. Wang, M. Zhou and L. Sun, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Applied Mathematics Letters, Vol. 21, No. 10, 2008, pp. 10241028. doi:10.1016/j.aml.2007.10.020

[16] Y. Li, “Hyers-Ulam Stability of Linear Differential Equations,” Thai Journal of Mathematics, Vol. 8, No. 2, 2010, pp. 215-219.

[17] Y. Li and Y. Shen, “Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order, Internat,” Journal of Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852.

[18] P. Gavruta, S. Jung and Y. Li, “Hyers-Ulam Stability for Second-Order Linear Differential Equations with Boundary Conditions,” Electron Journal of Differential Equations, Vol. 2011, No. 80, 2011, pp. 1-7.

[19] S. M. Jung, “Hyers-Ulam Stability of Linear Partial Differential Equations of First Order,” Applied Mathematics Letters, Vol. 22, No. 1, 2009, pp. 70-74. doi:10.1016/j.aml.2008.02.006

[20] M. E. Gordji, Y. J. Cho, M. B. Ghaemi and B. Alizadeh, “Stability of the Exact Second Order Partial Differential Equations,” Journal of Inequalities and Applications, Vol. 2011, 2011, Article ID: 306275. doi:10.1186/1029-242X-2011-81

[21] N. Lungu and C. Craciun, “Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation,” ISRN Mathematical Analysis, Vol. 2012, 2012, Article ID: 609754.

[22] L. C. Evans, “Partial Differential Equations,” 2nd Edition, Graduate Studies in Mathematics, V19 AMS, 2010.

[1] D. H. Hyers, “On the Stability of the Linear Functional Equation,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 27, No. 4, 1941, pp. 222-224. doi:10.1073/pnas.27.4.222

[2] T. M. Rassias, “On the Stability of the Linear Mapping in Banach Spaces,” Proceedings of the American Mathematical Society, Vol. 72, No. 2, 1978, pp. 297-300. doi:10.1090/S0002-9939-1978-0507327-1

[3] P. Gavruta, “A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 184, No. 3, 1994, pp. 431-436. doi:10.1006/jmaa.1994.1211

[4] S. M. Jung, “On the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 1, 1996, pp. 221-226. doi:10.1006/jmaa.1996.0433

[5] S. M. Jung, “Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis,” Hadronic Press, Palm Harbor, 2001.

[6] T. Miura, S.-E. Takahasi and H. Choda, “On the HyersUlam Stability of Real Continuous Function Valued Differentiable Map,” Tokyo Journal of Mathematics, Vol. 24, No. 2, 2001, pp. 467-476. doi:10.3836/tjm/1255958187

[7] C. G. Park, “On the Stability of the Linear Mapping in Banach Modules,” Journal of Mathematical Analysis and Applications, Vol. 275, 2002, pp. 711-720. doi:10.1016/S0022-247X(02)00386-4

[8] C. G. Park, “Homomorphisms between Poisson JC*-Algebras,” Bulletin of the Brazilian Mathematical Society, Vol. 36, No. 1, 2005, pp. 79-97. doi:10.1007/s00574-005-0029-z

[9] C. G. Park, Y.-S. Cho and M. Han, “Functional Inequalities Associated with Jordan-Von Neumanntype Additive Functional Equations,” Journal of Inequalities and Applications, Vol. 2007, 2007, Article ID: 41820.

[10] K.-W. Jun and Y.-H. Lee, “A Generalization of the HyersUlam-Rassias Stability of the Pexiderized Quadratice Quations,” Journal of Mathematical Analysis and Applications, Vol. 297, No. 1, 2004, pp. 70-86. doi:10.1016/j.jmaa.2004.04.009

[11] C. Alsina and R. Ger, “On Some Inequalities and Stability Results Related to the Exponential Function,” Journal of Inequalities and Applications, Vol. 2, No. 4, 1998, pp. 373-380.

[12] E. Takahasi, T. Miura and S. Miyajima, “On the HyersUlam Stability of the Banach Space-Valued Differential Equation ,” Bulletin of the Korean Mathematical Society, Vol. 39, No. 2, 2002, pp. 309-315. doi:10.4134/BKMS.2002.39.2.309

[13] T. Miura, S. Miyajima and S.-E. Takahasi, “A Characterization of Hyers-Ulam Stability of First Order Linear Differential Operators,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 1, 2007, pp. 136-146. doi:10.1016/S0022-247X(03)00458-X

[14] S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 1, 2005, pp. 139146. doi:10.1016/j.jmaa.2005.02.025

[15] G. Wang, M. Zhou and L. Sun, “Hyers-Ulam Stability of Linear Differential Equations of First Order,” Applied Mathematics Letters, Vol. 21, No. 10, 2008, pp. 10241028. doi:10.1016/j.aml.2007.10.020

[16] Y. Li, “Hyers-Ulam Stability of Linear Differential Equations,” Thai Journal of Mathematics, Vol. 8, No. 2, 2010, pp. 215-219.

[17] Y. Li and Y. Shen, “Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order, Internat,” Journal of Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852.

[18] P. Gavruta, S. Jung and Y. Li, “Hyers-Ulam Stability for Second-Order Linear Differential Equations with Boundary Conditions,” Electron Journal of Differential Equations, Vol. 2011, No. 80, 2011, pp. 1-7.

[19] S. M. Jung, “Hyers-Ulam Stability of Linear Partial Differential Equations of First Order,” Applied Mathematics Letters, Vol. 22, No. 1, 2009, pp. 70-74. doi:10.1016/j.aml.2008.02.006

[20] M. E. Gordji, Y. J. Cho, M. B. Ghaemi and B. Alizadeh, “Stability of the Exact Second Order Partial Differential Equations,” Journal of Inequalities and Applications, Vol. 2011, 2011, Article ID: 306275. doi:10.1186/1029-242X-2011-81

[21] N. Lungu and C. Craciun, “Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation,” ISRN Mathematical Analysis, Vol. 2012, 2012, Article ID: 609754.

[22] L. C. Evans, “Partial Differential Equations,” 2nd Edition, Graduate Studies in Mathematics, V19 AMS, 2010.