Previous Article in Journal
Fractional-Order Soft-Voting Ensemble Framework for Cohort-Level Gallstone Risk Prediction
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
MCA
Volume 31
Issue 5
10.3390/mca31050207
mca-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Some Salient Features of a (31)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws+

by
Yanga Gaxela
1,
Abdullahi Rashid Adem
1,
Ben Muatjetjeja
1,2,3,
Sivenathi Oscar Mbusi
3,
Ahmed H. Arnous
4,5 and
Anjan Biswas
6,7,8,9,10,*
1
Department of Mathematical Sciences, University of South Africa (Unisa), P.O. Box 392, Pretoria 0003, South Africa
2
Department of Mathematics, Faculty of Science, University of Botswana, Private Bag 22, Gaborone, Botswana
3
School of Mathematical and Statistical Sciences, North-West University, Private Bag X1290, Potchefstroom 2520, South Africa
4
Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai 602105, Tamil Nadu, India
5
Jadara Research Center, Jadara University, Irbid 21110, Jordan
6
Department of Mathematics & Physics, Grambling State University, Grambling, LA 71245-2715, USA
7
Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon 61080, Türkiye
8
Department of Physics and Electronics, Khazar University, Baku AZ1096, Azerbaijan
9
Applied Science Research Center, Applied Science Private University, Amman 11937, Jordan
10
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University (Medunsa), Pretoria 0204, South Africa
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2026, 31(5), 207; https://doi.org/10.3390/mca31050207 (registering DOI)
Submission received: 5 September 2026 / Revised: 26 September 2026 / Accepted: 29 September 2026 / Published: 1 October 2026
Versions Notes

Abstract

A generalized (3+1)-dimensional Hirota-type nonlinear wave equation is investigated by combining Lie point symmetry analysis, invariant reductions, exact traveling-wave construction, and the direct multiplier method. A potential formulation is introduced under an explicit decay condition that makes the nonlocal term well defined. The admitted point symmetries are re-derived from the prolonged invariance criterion, and several symmetry reductions are obtained, including mixed translations, a transverse shear, scaling, and a general traveling-wave phase. The reduced traveling-wave equation admits an affine Riccati representation with hyperbolic, trigonometric, rational, and Bernoulli-type branches under explicit parameter restrictions. The corresponding 3D, 2D, and density visualizations are retained to illustrate the resulting wave profiles. Zeroth-order multipliers are then used to construct local conserved vectors in divergence form. The analysis provides a verified symmetry and conservation-law framework for the proposed higher-dimensional Hirota-type extension and supplies exact solutions that can be used as analytical benchmarks. The transverse-variable reductions and the zeroth-order multiplier family are derived specifically for the present (3+1)-dimensional potential equation, while the Riccati functional forms are used as standard representations rather than claimed as new special functions.
Keywords: generalized (3+1)-dimensional Hirota-type equation; Lie point symmetries; group-invariant solutions; traveling-wave solutions; conserved vectors; Riccati method generalized (3+1)-dimensional Hirota-type equation; Lie point symmetries; group-invariant solutions; traveling-wave solutions; conserved vectors; Riccati method

Share and Cite

MDPI and ACS Style

Gaxela, Y.; Rashid Adem, A.; Muatjetjeja, B.; Mbusi, S.O.; Arnous, A.H.; Biswas, A. Some Salient Features of a (31)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws+. Math. Comput. Appl. 2026, 31, 207. https://doi.org/10.3390/mca31050207

AMA Style

Gaxela Y, Rashid Adem A, Muatjetjeja B, Mbusi SO, Arnous AH, Biswas A. Some Salient Features of a (31)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws+. Mathematical and Computational Applications. 2026; 31(5):207. https://doi.org/10.3390/mca31050207

Chicago/Turabian Style

Gaxela, Yanga, Abdullahi Rashid Adem, Ben Muatjetjeja, Sivenathi Oscar Mbusi, Ahmed H. Arnous, and Anjan Biswas. 2026. "Some Salient Features of a (31)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws+" Mathematical and Computational Applications 31, no. 5: 207. https://doi.org/10.3390/mca31050207

APA Style

Gaxela, Y., Rashid Adem, A., Muatjetjeja, B., Mbusi, S. O., Arnous, A. H., & Biswas, A. (2026). Some Salient Features of a (31)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws+. Mathematical and Computational Applications, 31(5), 207. https://doi.org/10.3390/mca31050207

Article Metrics

Article metric data becomes available approximately 24 hours after publication online.
Back to TopTop