Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type
Affiliation(s)
Tohoku University, Sendai, Japan. College of Engineering, Nihon University, Koriyama, Japan.
Tohoku University, Sendai, Japan. College of Engineering, Nihon University, Koriyama, Japan.
ABSTRACT
In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function
on 
, when 
is locally integrable on 
, and the integral 
converges. We now discard the last condition that 
should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.
In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function
on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.
Cite this paper
T. Morita and K. Sato, "Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 26-36. doi: 10.4236/am.2013.411A1005.
T. Morita and K. Sato, "Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 26-36. doi: 10.4236/am.2013.411A1005.
References
[1] K. Yosida, “The Algebraic Derivative and Laplace’s Differential Equation,” Proceedings of the Japan Academy, Vol. 59, Ser. A, 1983, pp. 1-4. [2] K. Yosida, “Operational Calculus,” Springer-Verlag, New York, 1982, Chapter VII. [3] J. Mikusiński, “Operational Calculus,” Pergamon Press, London, 1959. [4] T. Morita and K. Sato, “Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type,” Applied Mathematics, Vol. 4, No. 11A, 2013, pp. 13-21. [5] T. Morita and K. Sato, “Solution of Fractional Differential Equation in Terms of Distribution Theory,” Interdisciplinary Information Sciences, Vol. 12, No. 2, 2006, pp. 71-83. [6] T. Morita and K. Sato, “Neumann-Series Solution of Fractional Differential Equation,” Interdisciplinary Information Sciences, Vol. 16, No. 1, 2010, pp. 127-137. [7] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” Dover Publ., Inc., New York, 1972, Chapter 13. [8] M. Magnus and F. Oberhettinger, “Formulas and Theorems for the Functions of Mathematical Physics,” Chelsea Publ. Co., New York, 1949, Chapter VI. [9] T. Morita and K. Sato, “Liouville and Riemann-Liouville Fractional Derivatives via Contour Integrals,” Fractional Calculus and Applied Analysis, Vol. 16, No. 3, 2013, pp. 630-653. [10] L. Levine and R. Maleh, “Polynomial Solutions of the Classical Equations of Hermite, Legendre and Chebyshev,” International Journal of Mathematical Education in Science and Technology, Vol. 34, 2003, pp. 95-103. [11] F. Riesz and B. Sz.-Nagy, “Functional Analysis,” Dover Publ., Inc., New York, 1990, p. 146.
[1] K. Yosida, “The Algebraic Derivative and Laplace’s Differential Equation,” Proceedings of the Japan Academy, Vol. 59, Ser. A, 1983, pp. 1-4. [2] K. Yosida, “Operational Calculus,” Springer-Verlag, New York, 1982, Chapter VII. [3] J. Mikusiński, “Operational Calculus,” Pergamon Press, London, 1959. [4] T. Morita and K. Sato, “Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type,” Applied Mathematics, Vol. 4, No. 11A, 2013, pp. 13-21. [5] T. Morita and K. Sato, “Solution of Fractional Differential Equation in Terms of Distribution Theory,” Interdisciplinary Information Sciences, Vol. 12, No. 2, 2006, pp. 71-83. [6] T. Morita and K. Sato, “Neumann-Series Solution of Fractional Differential Equation,” Interdisciplinary Information Sciences, Vol. 16, No. 1, 2010, pp. 127-137. [7] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” Dover Publ., Inc., New York, 1972, Chapter 13. [8] M. Magnus and F. Oberhettinger, “Formulas and Theorems for the Functions of Mathematical Physics,” Chelsea Publ. Co., New York, 1949, Chapter VI. [9] T. Morita and K. Sato, “Liouville and Riemann-Liouville Fractional Derivatives via Contour Integrals,” Fractional Calculus and Applied Analysis, Vol. 16, No. 3, 2013, pp. 630-653. [10] L. Levine and R. Maleh, “Polynomial Solutions of the Classical Equations of Hermite, Legendre and Chebyshev,” International Journal of Mathematical Education in Science and Technology, Vol. 34, 2003, pp. 95-103. [11] F. Riesz and B. Sz.-Nagy, “Functional Analysis,” Dover Publ., Inc., New York, 1990, p. 146.