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Symmetry
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Symmetry and Its Applications in Partial Differential Equations
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Symmetry and Its Applications in Partial Differential Equations

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

Deadline for manuscript submissions: 28 February 2026 | Viewed by 13165

Special Issue Editors

Dr. Mujahid Iqbal

E-Mail Website
Guest Editor
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
Interests: mathematical physics; partial differential equations; nonlinear PDEs; fractional calculus; exact traveling and solitary wave solutions
Prof. Dr. Dianchen Lu

E-Mail Website
Guest Editor
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
Interests: soliton theory; nonlinear system; bifurcation analysis; homotopy analysis method; numerical analysis; mathematical physics; partial differential equations; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Partial differential equations have become a useful tool to describe the natural phenomena of science and engineering. Nonlinear partial differential equations (NLPDEs) arise in many branches of science such as mathematics, physics, mechanics, water waves, computational fluid dynamics, optics, quantum mechanics, shallow water, and engineering.

NLPDEs are widely used to describe physical phenomena in natural science, such as plasma physics, optical fibers, biology, solid-state physics, fluid dynamics, and play a crucial role in research in many disciplines, including in the concept of symmetry. On the other hand, the symmetric properties of NLPDEs are of great importance for the solution of problems in many areas of mathematics. The role of symmetry has also proven to be fundamental in other different disciplines, such as biology, chemistry, and psychology. In this Special Issue, this correlation will be in the foreground. Solutions of NLPDEs play an important role in understanding the mechanisms of many physical phenomena and processes in various areas of natural science. They can help to analyze the stability of these solutions and the movement role of the wave by making graphs of the exact solutions.

Potential themes of interest in this research topic include, but are not limited to, the following:

  • Nonlinear partial differential equations;
  • Fractional differential equations;
  • Soliton wave theory;
  • Mathematical methods;
  • Stability analysis of dynamical systems;
  • Nonlinear water waves;
  • Computational fluid dynamics;
  • Fiber optics;
  • Fractional integral inequalities;
  • Computational quantum mechanics;
  • Ion-acoustic waves;
  • Nonlinear plasma models.

We thus call researchers to contribute to this new Special Issue of the journal Symmetry, titled “Symmetry and Its Applications in Partial Differential Equations”, via the MDPI submission system. We look forward to receiving your contributions of review and original research articles that deal with recent topics and advances in partial differential equations and symmetry. The published papers in this Special Issue of Symmetry could provide crucial examples and new possible research directions for further advancements.

Dr. Mujahid Iqbal
Prof. Dr. Dianchen Lu
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

  • partial differential equations
  • mathematical methods
  • exact solutions
  • solitons
  • solitary waves
  • analytical and numerical wave solutions

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (11 papers)

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Research

29 pages, 2347 KB  
Article
The Hamiltonian Form of the KdV Equation: Multiperiodic Solutions and Applications to Quantum Mechanics
by Alfred R. Osborne and Uggo Ferreira de Pinho
Symmetry 2025, 17(12), 2015; https://doi.org/10.3390/sym17122015 - 21 Nov 2025
Viewed by 199
Abstract
In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly [...] Read more.
In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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26 pages, 2582 KB  
Article
Lie Symmetry Analysis, Optimal Systems and Physical Interpretation of Solutions for the KdV-Burgers Equation
by Faiza Afzal and Alina Alb Lupas
Symmetry 2025, 17(11), 1981; https://doi.org/10.3390/sym17111981 - 16 Nov 2025
Viewed by 215
Abstract
This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to [...] Read more.
This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than 0.5% in the Burgers limit and 3.2% in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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24 pages, 4021 KB  
Article
A Modified Analytical Data-Mapping Framework for Symmetric Multiscale Soliton and Chaotic Dynamics
by Syeda Sarwat Kazmi, Muhammad Bilal Riaz and Faisal Z. Duraihem
Symmetry 2025, 17(11), 1963; https://doi.org/10.3390/sym17111963 - 14 Nov 2025
Viewed by 299
Abstract
The (3 + 1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, a model that describes long-wave interactions and has numerous applications in mathematics, engineering, and physics, is examined in this work. First, a wave transformation is used to reduce the equation to lower dimensions. The modified Khater method [...] Read more.
The (3 + 1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, a model that describes long-wave interactions and has numerous applications in mathematics, engineering, and physics, is examined in this work. First, a wave transformation is used to reduce the equation to lower dimensions. The modified Khater method is then used to derive different types of solitary wave solutions, such as chirped, kink, periodic, and kink-bright types. By allocating suitable constant parameters, 3D, 2D, and contour plots are created to demonstrate the physical behavior of these solutions. Phase portraits are used to qualitatively analyze the undisturbed planar system using bifurcation theory. The system is then perturbed by an external force, resulting in chaotic dynamics. Chaos in the system is confirmed using multiple diagnostic tools, including time series plots, Poincaré sections, chaotic attractors, return maps, bifurcation diagrams, power spectra, and Lyapunov exponents. The stability of the model is further investigated with varying initial conditions. A bidirectional scatter plot technique, which efficiently reveals overlapping regions using data point distributions, is presented for comparing solution behaviors. Overall, this work offers useful tools for advancing applied mathematics research as well as a deeper understanding of nonlinear wave dynamics. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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29 pages, 1831 KB  
Article
On the Performance of Physics-Based Neural Networks for Symmetric and Asymmetric Domains: A Comparative Study and Hyperparameter Analysis
by Rafał Brociek, Mariusz Pleszczyński and Dawood Asghar Mughal
Symmetry 2025, 17(10), 1698; https://doi.org/10.3390/sym17101698 - 10 Oct 2025
Cited by 1 | Viewed by 724
Abstract
This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge [...] Read more.
This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge with the approximation capabilities of neural networks, allowing the modeling of physical phenomena without explicit domain discretization. In addition to evaluating accuracy against analytical solutions (where available) and established numerical methods, the study systematically examines the impact of key hyperparameters—such as the number of hidden layers, neurons per layer, and training points—on solution quality and stability. The impact of a symmetric domain on solution speed is also analyzed. The experimental results highlight the strengths and limitations of PINNs and provide practical guidelines for their effective application as an alternative or complement to traditional computational approaches. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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16 pages, 13982 KB  
Article
Exploring Chaos in Fractional Order Systems: A Study of Constant and Variable-Order Dynamics
by Reem Allogmany, Nada A. Almuallem, Reima Daher Alsemiry and Mohamed A. Abdoon
Symmetry 2025, 17(4), 605; https://doi.org/10.3390/sym17040605 - 16 Apr 2025
Cited by 4 | Viewed by 1120
Abstract
Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. [...] Read more.
Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. Out of these, fractional jerk chaotic systems have gained attention regarding their applications in secure communication, signal processing, and control systems. This work develops a comparative analysis of a fractional jerk system that includes constant- and variable-order derivatives to contribute to chaos–stability analysis. Additionally, this study uncovers novel chaotic behaviors, further expanding our understanding of complex dynamical systems. The results yield new insights into using variable-order dynamics to enable chaotic systems to better adapt to real applications. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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14 pages, 1070 KB  
Article
Efficient Numerical Techniques for Investigating Chaotic Behavior in the Fractional-Order Inverted Rössler System
by Mohamed Elbadri, Dalal M. AlMutairi, D. K. Almutairi, Abdelgabar Adam Hassan, Walid Hdidi and Mohamed A. Abdoon
Symmetry 2025, 17(3), 451; https://doi.org/10.3390/sym17030451 - 18 Mar 2025
Cited by 3 | Viewed by 863
Abstract
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted [...] Read more.
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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19 pages, 6089 KB  
Article
Symmetry Breaking in Fractional Difference Chaotic Equations and Their Control
by Louiza Diabi, Adel Ouannas, Giuseppe Grassi and Shaher Momani
Symmetry 2025, 17(3), 352; https://doi.org/10.3390/sym17030352 - 26 Feb 2025
Cited by 3 | Viewed by 752
Abstract
This manuscript presents new fractional difference equations; we investigate their behaviors in-depth in commensurate and incommensurate order cases. The work exploits a range of numerical approaches involving bifurcation, the Maximum Lyapunov exponent (LEm), and the visualization of phase portraits and also uses the [...] Read more.
This manuscript presents new fractional difference equations; we investigate their behaviors in-depth in commensurate and incommensurate order cases. The work exploits a range of numerical approaches involving bifurcation, the Maximum Lyapunov exponent (LEm), and the visualization of phase portraits and also uses the C0 complexity algorithm and the approximation entropy ApEn to evaluate the intricacy and verify the chaotic features. Thus, the outcomes indicate that the suggested fractional-order map can display a variety of hidden attractors and symmetry breaking if it has no fixed points. Additionally, nonlinear controllers are offered to stabilize the fractional difference equations. As a result, the study highlights how the map’s sensitivity to the fractional derivative parameters produces different dynamics. Lastly, simulations using MATLAB R2024b are run to validate the results. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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22 pages, 580 KB  
Article
Identification of Boundary Conditions in a Spherical Heat Conduction Transmission Problem
by Miglena N. Koleva and Lubin G. Vulkov
Symmetry 2024, 16(11), 1507; https://doi.org/10.3390/sym16111507 - 10 Nov 2024
Cited by 1 | Viewed by 1805
Abstract
Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat [...] Read more.
Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat equations with spherical symmetry. The zero degeneracy at the left interval 0<r<R1 leads to solution difficulties in the one-dimensional interface problem. So, we first investigate the well-posedness of the direct (forward) problem in special weighted Sobolev spaces. Then, we formulate three groups of unknown boundary conditions and inverse problems upon internal point measurements for the heat equation with spherical symmetry. Second-order finite difference scheme approaches for direct and inverse problems are developed. Computational test examples illustrate the theoretical statements proposed. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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13 pages, 306 KB  
Article
Lie Symmetry Analysis and Explicit Solutions to the Estevez–Mansfield–Clarkson Equation
by Aliyu Isa Aliyu, Jibrin Sale Yusuf, Malik Muhammad Nauman, Dilber Uzun Ozsahin, Baba Galadima Agaie, Juliana Haji Zaini and Huzaifa Umar
Symmetry 2024, 16(9), 1194; https://doi.org/10.3390/sym16091194 - 11 Sep 2024
Cited by 5 | Viewed by 1696
Abstract
In this study, we investigate the symmetry analysis and explicit solutions for the Estevez–Mansfield–Clarkson (EMC) equation. Our main objectives are to identify the Lie point symmetries of the EMC equation, construct an optimal system of one-dimensional subalgebras, and reduce the EMC equation to [...] Read more.
In this study, we investigate the symmetry analysis and explicit solutions for the Estevez–Mansfield–Clarkson (EMC) equation. Our main objectives are to identify the Lie point symmetries of the EMC equation, construct an optimal system of one-dimensional subalgebras, and reduce the EMC equation to a set of ordinary differential equations (ODEs). We employ the Riccati–Bernoulli sub-ODE method (RBSODE) to solve these reduced ODEs and obtain explicit solutions for the EMC model. The obtained solutions are validated using numerical analyses, and corresponding figures are presented to illustrate the physical implications of the derived solutions. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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23 pages, 5305 KB  
Article
An Analytical Study of the Mikhailov–Novikov–Wang Equation with Stability and Modulation Instability Analysis in Industrial Engineering via Multiple Methods
by Md Nur Hossain, M. Mamun Miah, M. S. Abbas, K. El-Rashidy, J. R. M. Borhan and Mohammad Kanan
Symmetry 2024, 16(7), 879; https://doi.org/10.3390/sym16070879 - 11 Jul 2024
Cited by 9 | Viewed by 2073
Abstract
Solitary waves, inherent in nonlinear wave equations, manifest across various physical systems like water waves, optical fibers, and plasma waves. In this study, we present this type of wave solution within the integrable Mikhailov–Novikov–Wang (MNW) equation, an integrable system known for representing localized [...] Read more.
Solitary waves, inherent in nonlinear wave equations, manifest across various physical systems like water waves, optical fibers, and plasma waves. In this study, we present this type of wave solution within the integrable Mikhailov–Novikov–Wang (MNW) equation, an integrable system known for representing localized disturbances that persist without dispersing, retaining their form and coherence over extended distances, thereby playing a pivotal role in understanding nonlinear dynamics and wave phenomena. Beyond this innovative work, we examine the stability and modulation instability of its gained solutions. These new solitary wave solutions have potential applications in telecommunications, spectroscopy, imaging, signal processing, and pulse modeling, as well as in economic systems and markets. To derive these solitary wave solutions, we employ two effective methods: the improved Sardar subequation method and the (℧′/℧, 1/℧) method. Through these methods, we develop a diverse array of waveforms, including hyperbolic, trigonometric, and rational functions. We thoroughly validated our results using Mathematica software to ensure their accuracy. Vigorous graphical representations showcase a variety of soliton patterns, including dark, singular, kink, anti-kink, and hyperbolic-shaped patterns. These findings highlight the effectiveness of these methods in showing novel solutions. The utilization of these methods significantly contributes to the derivation of novel soliton solutions for the MNW equation, holding promise for diverse applications throughout different scientific domains. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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16 pages, 2108 KB  
Article
Semi-Separable Potentials as Solutions to the 3D Inverse Problem of Newtonian Dynamics
by Thomas Kotoulas
Symmetry 2024, 16(2), 198; https://doi.org/10.3390/sym16020198 - 7 Feb 2024
Cited by 1 | Viewed by 1433
Abstract
We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z) [...] Read more.
We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately. Full article
(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)
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