Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform

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

We discuss the solution of Laplace’s differential equation and a fractional differential equation of that type, by using analytic continuations of Riemann-Liouville fractional derivative and of Laplace transform. We show that the solutions, which are obtained by using operational calculus in the framework of distribution theory in our preceding papers, are obtained also by the present method.

KEYWORDS

Laplace’s Differential Equation, Kummer’s Differential Equation, Fractional Differential Equation, Laplace Transform, Analytic Continuation via Hankel’s Contour

Laplace’s Differential Equation, Kummer’s Differential Equation, Fractional Differential Equation, Laplace Transform, Analytic Continuation via Hankel’s Contour

Cite this paper

Morita, T. and Sato, K. (2014) Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform.*Applied Mathematics*, **5**, 1229-1239. doi: 10.4236/am.2014.58115.

Morita, T. and Sato, K. (2014) Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform.

References

[1] Yosida, K. (1983) The Algebraic Derivative and Laplace’s Differential Equation. Proceedings of Japan Academy, 59, 14.

http://dx.doi.org/10.3792/pjaa.59.1

[2] Yosida, K. (1982) Operational Calculus. SpringerVerlag, New York, Chapter VII.

[3] Morita, T. and Sato, K. (2013) Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type. Applied Mathematics, 4, 1321.

http://dx.doi.org/10.4236/am.2013.411A1003

[4] Morita, T. and Sato, K. (2013) Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type. Applied Mathematics, 4, 2636.

http://dx.doi.org/10.4236/am.2013.411A1005

[5] Mikusiński, J. (1959) Operational Calculus. Pergamon Press, London.

[6] Morita, T. and Sato, K. (2006) Solution of Fractional Differential Equation in Terms of Distribution Theory. Interdisciplinary Information Sciences, 12, 7183.

[7] Morita, T. and Sato, K. (2010) NeumannSeries Solution of Fractional Differential Equation. Interdisciplinary Information Sciences, 16, 127137.

http://dx.doi.org/10.4036/iis.2010.127

[8] Morita, T. and Sato, K. (2013) Liouville and RiemannLiouville Fractional Derivatives via Contour Integrals. Fractional Calculus and Applied Analysis, 16, 630653.

http://dx.doi.org/10.2478/s1354001300409

[9] Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publ., Inc., New York, Chapter 13.

[10] Magnus, M. and Oberhettinger, F. (1949) Formulas and Theorems for the Functions of Mathematical Physics. Chelsea Publ. Co., New York, Chapter VI.

[11] Whittaker, E.T. and Watson, G.N. (1935) A Course of Modern Analysis. Cambridge U.P., Cambridge

[1] Yosida, K. (1983) The Algebraic Derivative and Laplace’s Differential Equation. Proceedings of Japan Academy, 59, 14.

http://dx.doi.org/10.3792/pjaa.59.1

[2] Yosida, K. (1982) Operational Calculus. SpringerVerlag, New York, Chapter VII.

[3] Morita, T. and Sato, K. (2013) Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type. Applied Mathematics, 4, 1321.

http://dx.doi.org/10.4236/am.2013.411A1003

[4] Morita, T. and Sato, K. (2013) Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type. Applied Mathematics, 4, 2636.

http://dx.doi.org/10.4236/am.2013.411A1005

[5] Mikusiński, J. (1959) Operational Calculus. Pergamon Press, London.

[6] Morita, T. and Sato, K. (2006) Solution of Fractional Differential Equation in Terms of Distribution Theory. Interdisciplinary Information Sciences, 12, 7183.

[7] Morita, T. and Sato, K. (2010) NeumannSeries Solution of Fractional Differential Equation. Interdisciplinary Information Sciences, 16, 127137.

http://dx.doi.org/10.4036/iis.2010.127

[8] Morita, T. and Sato, K. (2013) Liouville and RiemannLiouville Fractional Derivatives via Contour Integrals. Fractional Calculus and Applied Analysis, 16, 630653.

http://dx.doi.org/10.2478/s1354001300409

[9] Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publ., Inc., New York, Chapter 13.

[10] Magnus, M. and Oberhettinger, F. (1949) Formulas and Theorems for the Functions of Mathematical Physics. Chelsea Publ. Co., New York, Chapter VI.

[11] Whittaker, E.T. and Watson, G.N. (1935) A Course of Modern Analysis. Cambridge U.P., Cambridge