On the relationship between the inhomogeneous wave and Helmholtz equations in a fractional setting. (English) Zbl 1474.35645

Summary: We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.


35R11 Fractional partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text: DOI


[1] Tikhonov, A. N.; Samarskii, A. A., O printsipe izlucheniya [On the radiation principle], Journal of Experimental and Theoretical Physics, 18, 2, 243-248 (1948)
[2] Vainberg, B. R., Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations, (Russian Mathematical Surveys, vol. 21, no. 3, pp. 115-193, 1966) · Zbl 0172.13703
[3] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4, 2, 153-192 (2001) · Zbl 1054.35156
[4] Duan, J.-S., The periodic solution of fractional oscillation equation with periodic input, Advances in Mathematical Physics, 2013 (2013) · Zbl 1291.34008
[5] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North Holland Mathematical Studies v.204 (2006), New York, NY, USA: Elsevier, New York, NY, USA · Zbl 1092.45003
[6] Turmetov, B. K.; Torebek, B. T., On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation, New York Journal of Mathematics, 20, 1237-1251 (2014) · Zbl 1316.35293
[7] Saxena, R.; Tomovski, Z.; Sandev, T., Fractional Helmholtz and fractional wave equations with Riesz-Feller and generalized Riemann-Liouville fractional derivatives, European Journal of Pure and Applied Mathematics, 7, 3, 312-334 (2014) · Zbl 1389.35312
[8] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), Academic Press · Zbl 0918.34010
[9] Barrett, J. H., Differential equations of non-integer order, Canadian Journal of Mathematics, 6, 529-541 (1954) · Zbl 0058.10702
[10] Grigoryan, V., Partial Differential Equations. Partial Differential Equations, Math 124A - fall 2010 lecture notes (2010)
[11] Candelpergher, B.; Nosmas, J.-C.; Pham, F., Approche de la Résurgence (1993), Paris: Actualités Mathématiques - Hermann, Paris · Zbl 0791.32001
[12] Wong, R., Asymptotic Approximations of Integrals (1989), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0679.41001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.