GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION

Abdelmjid Benmerrous; Lalla Saadia Chadli; M'hamed Elomari

GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION

Abstract

This paper focuses on the fractional wave problem with the use of a new fractional derivative in Colombeau algebra. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, the existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

Keywords

References

[1] Baz, Z., Helvaci, M., Ikiz, T., Veliev, E. I. (2024), Integral Equations For The Problem Of Wave Equations For The Problem Of Wave Diffraction On A Flat Strip: Alternative Representation. TWMS Journal of Applied and Engineering Mathematics, V.14, N.1, 113-122.
[2] Benmerrous, A., Bourhim, F. E., El Mfadel, A., Elomari, M. H., (2024), Solving a time-fractional semilinear hyperbolic equations by Fourier truncation with boundary conditions. Chaos, Solitons & Fractals, 185, 115086.
[3] Benmerrous, A., Chadli, L. S., Elomari, M. H. (2026). Generalized fractional heat equation in extended Colombeau algebras. TWMS Journal of Applied and Engineering Mathematics, 16(1), 16-31.
[4] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M., and Melliani, S., (2024), Conformable cosine family and nonlinear fractional differential equations. Filomat, 38(9), 3193-3206.
[5] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M. H., and Melliani, S., (2022), Generalized Cosine Family, Journal of Elliptic and Parabolic Equations, 8(1), pp. 367-381.
[6] Benmerrous, A., Chadli, L. s., Moujahid, A., Elomari, M. H., and Melliani, S., (2023), Generalized Fractional Cosine Family, International Journal of Difference Equations (IJDE), 18(1), pp. 11-34.
[7] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M. H., and Melliani, S., (2024), Generalized solutions for time ψ-fractional evolution equations, Boletim da Sociedade Paranaense de Matematica, 42, 1-14.
[8] Benmerrous, A., Chadli, L. S., Moujahid, A., and Melliani, S., (2023), Generalized solutions for time ψ-fractional heat equation., Filomat, 37(27), 9327-9337.

[9] Benmerrous, A., Chadli, L. S., Moujahid, A., and Melliani, S., (2024), Generalized solutions for fractional Schrödinger equation., TWMS Journal of Applied and Engineering Mathematics, V.14, N.4, 1361-1373
[10] Benmerrous, A., Chadli, L. S., Moujahid, A., M’hamed, E., Melliani, S., (2022), Generalized solution of Schrödinger equation with singular potential and initial data, Int. J. Nonlinear Anal. Appl, 13(1), pp. 3093-3101.
[11] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M. H., and Melliani, S., (2024), On a fractional Cauchy problem with singular initial data, Nonautonomous Dynamical Systems, 11(1), 20240004.
[12] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M. H., and Melliani, S., (2022, October), Solution of Schrödinger type Problem in Extended Colombeau Algebras, In 2022 8th International Conference on Optimization and Applications (ICOA), pp. 1-5.
[13] Benmerrous, A., Chadli, L. S., Moujahid, A., Elomari, M. H., and Melliani, S., (2023), Solution of nonhomogeneous wave equation in extended Colombeau algebras, International Journal of Difference Equations (IJDE), 18(1), 107-118.
[14] Benmerrous, A., Elomari, M. H., and El mfadel, A. (2026), Solving the Fractional Schrödinger Equation with Singular Potential by Means of the Fourier Transform, Kragujevac Journal of Mathematics, 50(6), 921-929.
[15] Biagioni, H. A., (1990), A nonlinear theory of generalized functions. Springer, Berlin-Hedelberg-New York.
[16] Biagioni, H. A., Oberguggenberger, M., (1992), Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations. SIAM J. Math. Anal. 23, 923–940. doi:10.1137/0523049.
[17] Bourgain, J., (1999), Global solutions of nonlinear Schr¨odinger equations, AMS, Colloquium Publications, vol.46.
[18] Burtscher, A., Kunzinger, M., (2012), Algebras of generalized functions with smooth parameter dependence. Proc. Edinb. Math. Soc. 55, 105–124. doi:10.1017/s0013091510001410.
[19] Chadli, L. S., Benmerrous, A., Moujahid, A., Elomari, M. H., and Melliani, S., (2022), Generalized Solution of Transport Equation, In Recent Advances in Fuzzy Sets Theory, Fractional Calculus, Dynamic Systems and Optimization, pp. 101-111.
[20] Colombeau, J.F., (1985), Elementary introduction to new generalized functions, North-Holland Math. Stud. 113, North-Holland Publishing Co., Amsterdam.
[21] Debrouwere, A., Vernaeve, H., Vindas, J., (2018), Optimal embeddings of ultradistributions into differential algebras. Monatsh. Math. 186, 407–438. doi:10.1007/s00605-017-1066-6.
[22] El Mfadel, A., Bourhim, F. E., Elomari, M., (2022), Existence of mild solutions for semilinear ψ−Caputo-type fractional evolution equations with nonlocal conditions in Banach spaces. Results in Nonlinear Analysis, 5(4), 459-472.
23] Garetto, C., Hormann, G., (2005), Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc. 48, 603–629. doi:10.1017/0013091504000148.
[24] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R., (2001), Geometric theory of generalized functions with applications to general relativity, Mathematics and its Applications, 537. Kluwer Acad. Publ, Dordrecht. doi:10.1007/978-94-015-9845-35.
[25] Hermann, R., Oberguggenberger, M., (1999), Ordinary differential equations and generalized functions, in: Non linear Theory of Generalized Functions, Chapman and Hall, 85-98. doi:10.1201/9780203745458-8.
[26] Oberguggenberger, M., (1992), Multiplication of distributions and applications to partial differential equations, Pitman Res. Notes Math. Ser. 259, Longman, Harlow. doi:10.1137/1036076.
[27] Pazy, A., (1985), Semigroups of linear operators and applications to partial differential equations, Bull. Amer. Math. Soc.(N.S.) 12. doi:10.1007/978-1-4612-5561-17.
[28] Pilipovic, S., Scarpalezos, D., (2001), Regularity properties of distributions and ultradistributions. Proc. Amer. Math. Soc. 129, 3531–3537. doi:10.1090/s0002-9939-01-06013-0.
[29] Pilipovic, S., Scarpalezos, D., Valmorin, V., (2006), Equalities in algebras of generalized functions. Forum Math. 18, 789–802. doi:10.1515/forum.2006.039.
[30] Pilipovic, S., Scarpalezos, D., Vindas, J., (2013), Regularity properties of distributions through sequences of functions. Monatsh. Math. 170, 307–322. doi:10.1007/s00605-012-0410-0.
[31] Schwartz, L., (1966), Th´eorie des distributions. Hermann, Paris.
[32] Segal, I., (1963), Non-linear semi-groups, Ann. Math. 78, 339-364. doi:10.2307/1970347.
[33] Sova, M., (1966), cosine operator functions, Warszawa, 47.
[34] Stojanovic, M., (2012), Foundation of the fractional calculus in generalized function algebra. Analysis and Applications. 10, 439-467. doi:10.1142/s0219530512500212.
[35] Travis, C. C., Webb, G. F., (1978), Cosine families and abstract nonlinear second order differential equations. Acta Mathematica Hungarica. 1588-2632. doi:10.1007/bf01902205.

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems, Partial Differential Equations, Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

Abdelmjid Benmerrous ^*
0000-0001-9620-8161
Morocco

Lalla Saadia Chadli
0000-0002-0659-8774
Morocco

M'hamed Elomari
0000-0002-4256-1434
Morocco

Publication Date

May 4, 2026

Submission Date

April 9, 2025

Acceptance Date

September 2, 2025

Published in Issue

Year 2026 Volume: 16 Number: 5

IZ

https://izlik.org/JA78JT85XM

Cite

RIS / Bibtex

APA

Benmerrous, A., Chadli, L. S., & Elomari, M. (2026). GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION. TWMS Journal of Applied and Engineering Mathematics, 16(5), 563-575. https://izlik.org/JA78JT85XM

AMA

1.Benmerrous A, Chadli LS, Elomari M. GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION. JAEM. 2026;16(5):563-575. https://izlik.org/JA78JT85XM

Chicago

Benmerrous, Abdelmjid, Lalla Saadia Chadli, and M’hamed Elomari. 2026. “GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION”. TWMS Journal of Applied and Engineering Mathematics 16 (5): 563-75. https://izlik.org/JA78JT85XM.

EndNote

Benmerrous A, Chadli LS, Elomari M (May 1, 2026) GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION. TWMS Journal of Applied and Engineering Mathematics 16 5 563–575.

IEEE

[1]A. Benmerrous, L. S. Chadli, and M. Elomari, “GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION”, JAEM, vol. 16, no. 5, pp. 563–575, May 2026, [Online]. Available: https://izlik.org/JA78JT85XM

ISNAD

Benmerrous, Abdelmjid - Chadli, Lalla Saadia - Elomari, M’hamed. “GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION”. TWMS Journal of Applied and Engineering Mathematics 16/5 (May 1, 2026): 563-575. https://izlik.org/JA78JT85XM.

JAMA

1.Benmerrous A, Chadli LS, Elomari M. GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION. JAEM. 2026;16:563–575.

MLA

Benmerrous, Abdelmjid, et al. “GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 5, May 2026, pp. 563-75, https://izlik.org/JA78JT85XM.

Vancouver

1.Abdelmjid Benmerrous, Lalla Saadia Chadli, M’hamed Elomari. GENERALIZED SOLUTIONS FOR TIME $\psi$-FRACTIONAL WAVE EQUATION. JAEM [Internet]. 2026 May 1;16(5):563-75. Available from: https://izlik.org/JA78JT85XM