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Symmetry
Special Issues
Integral/Differential Equations and Symmetry
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Integral/Differential Equations and Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 15 April 2026 | Viewed by 837

Special Issue Editors

Prof. Dr. Yaqing Liu

E-Mail Website
Guest Editor
College of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China
Interests: integrable systems; mathematical physics; partial differential equations
Dr. Manwai Yuen

E-Mail Website
Guest Editor
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
Interests: partial differential equations; symmetry reduction; blowup; euler-poisson equations; euler equations with or without coriolis force; camassa-holm equations; navier-stokes equations; magnetohydrody-namics (MHD); analytical and exact solutions; mathematical methods in fluids; classical cosmology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleague, 

This Special Issue explores the significant and fruitful intersection of symmetry analysis with the theory of integral and differential equations. Symmetry, in its various forms—from Lie point symmetries and nonlocal symmetries to more abstract algebraic and geometric structures—provides a powerful and unifying framework for tackling nonlinear problems.

The collected works demonstrate how symmetry methods are not merely a tool for solving equations but are fundamental to understanding their intrinsic properties. Contributions highlight advancements in areas such as symmetry-based exact solutions, invariant discretization techniques, group classification, and the discovery of conservation laws. Furthermore, the issue showcases applications across diverse scientific fields, including mathematical physics, fluid dynamics, and biology, underscoring the universal utility of these approaches.

By bridging theoretical developments with practical applications, this issue aims to inspire further research and highlight symmetry as an indispensable principle in the modern analysis of complex dynamical systems.

Prof. Dr. Yaqing Liu
Dr. Manwai Yuen
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • lie symmetry analysis
  • differential equations
  • integral equations
  • symmetry methods
  • integrable systems
  • nonlinear waves
  • invariant solutions
  • conservation laws
  • nonlocal symmetries

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Published Papers (2 papers)

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Research

14 pages, 7094 KB  
Article
Stem Structures and Novel Localized Wave Molecules of the (2+1)-Dimensional pKP-BKP Equation
by Wei Zhu, Xueqing Zhang and Bo Ren
Symmetry 2025, 17(12), 2028; https://doi.org/10.3390/sym17122028 - 26 Nov 2025
Viewed by 255
Abstract
In this paper, asymptotic analysis for the (2+1)-dimensional potential Kadomtsev Petviashvili B-type Kadomtsev Petviashvili (pKP-BKP) equation is conducted to establish this limiting. The study examines the stem structures resulting from resonance collisions of the two-soliton solution in the (2+1)-dimensional pKP-BKP equation. The trajectories, [...] Read more.
In this paper, asymptotic analysis for the (2+1)-dimensional potential Kadomtsev Petviashvili B-type Kadomtsev Petviashvili (pKP-BKP) equation is conducted to establish this limiting. The study examines the stem structures resulting from resonance collisions of the two-soliton solution in the (2+1)-dimensional pKP-BKP equation. The trajectories, amplitudes and velocities of the soliton arms and the lengths of the stem structures are calculated through asymptotic analysis. By combining the characteristic line analysis and the velocity resonance mechanism, novel bound states consisting of transformed wave–soliton molecules and transformed wave molecules are systematically constructed. These results provide explicit insight into the formation and evolution of localized nonlinear waves in the pKP-BKP system, and they offer a concrete theoretical foundation for wave resonance and bound states in higher-dimensional dispersive media. Full article
(This article belongs to the Special Issue Integral/Differential Equations and Symmetry)
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32 pages, 2523 KB  
Article
Hybrid Nanofluid Flow and Heat Transfer in Inclined Porous Cylinders: A Coupled ANN and Numerical Investigation of MHD and Radiation Effects
by Muhammad Fawad Malik, Reem Abdullah Aljethi, Syed Asif Ali Shah and Sidra Yasmeen
Symmetry 2025, 17(11), 1998; https://doi.org/10.3390/sym17111998 - 18 Nov 2025
Viewed by 377
Abstract
This study investigates the thermal characteristics of two hybrid nanofluids, single-walled carbon nanotubes with titanium dioxide (SWCNT−TiO2) and multi-walled carbon nanotubes with copper (MWCNT−Cu [...] Read more.
This study investigates the thermal characteristics of two hybrid nanofluids, single-walled carbon nanotubes with titanium dioxide (SWCNT−TiO2) and multi-walled carbon nanotubes with copper (MWCNT−Cu), as they flow over an inclined, porous, and longitudinally stretched cylindrical surface with kerosene as the base fluid. The model takes into consideration all of the consequences of magnetohydrodynamic (MHD) effects, thermal radiation, and Arrhenius-like energy of activation. The outcomes of this investigation hold practical significance for energy storage systems, nuclear reactor heat exchangers, electronic cooling devices, biomedical hyperthermia treatments, oil and gas transport processes, and aerospace thermal protection technologies. The proposed hybrid ANN–numerical framework provides an effective strategy for optimizing the thermal performance of hybrid nanofluids in advanced thermal management and energy systems. A set of coupled ordinary differential equations is created by applying similarity transformations to the governing nonlinear partial differential equations that reflect conservation of mass, momentum, energy, and species concentration. The boundary value problem solver bvp4c, which is based in MATLAB (R2020b), is used to solve these equations numerically. The findings demonstrate that, in comparison to the MWCNT−Cu/kerosene nanofluid, the SWCNT−TiO2/kerosene hybrid nanofluid improves the heat transfer rate (Nusselt number) by up to 23.6%. When a magnetic field is applied, velocity magnitudes are reduced by almost 15%, and the temperature field is enhanced by around 12% when thermal radiation is applied. The impact of important dimensionless variables, such as the cylindrical surface’s inclination angle, the medium’s porosity, the magnetic field’s strength, the thermal radiation parameter, the curvature ratio, the activation energy, and the volume fraction of nanoparticles, is investigated in detail using a parametric study. According to the comparison findings, at the same flow and thermal boundary conditions, the SWCNT−TiO2/kerosene hybrid nanofluid performs better thermally than its MWCNT−Cu/kerosene counterpart. These results offer important new information for maximizing heat transfer in engineering systems with hybrid nanofluids and inclined porous geometries under intricate physical conditions. With its high degree of agreement with numerical results, the ANN model provides a computationally effective stand-in for real-time thermal system optimization. Full article
(This article belongs to the Special Issue Integral/Differential Equations and Symmetry)
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