Original Differential Equation Research Articles

Bifurcation on 4-dimensional Canards with Hyper Catastrophe

In 4-dimensional slow-fast system, under the condition of ”symmetry”, there exists ”structural stability”. It is, however, of one parameter for the slow vector. On other parameters for all slow/fast vectors, it is not yet discussed still now as it is very complicated geometrical structure. In the ”Hyper catastrophe on 4-dimensional canards”, it is confirmed due to the existence of ”bifurcation”, because ”catastrophe” is a bifurcation problem itself. In the beginning of catastrophe theory, the word ”structural stability” is used for the original differential equations and not be used for the multi variable functions. In the slow-fast system having canards, what kinds of structure are there? It is used for the parameter, which depends on the existence of canards. Through this paper, it will become clear.

Solution analysis of Solow Growth Model for financial practices and applications

Solow Growth Model (SGM) is an economic model that is exogenous in nature and examines the relationship between the output and input levels in an economy over a period of time. It projects long-term economic growth in relation to labour (population growth), savings rate, and technological development. However, traditional approaches to solving the Solow growth model may rely on complex mathematical techniques that might not give an accurate representation of real-world economic dynamics. Thus, this paper applies the Natural Decomposition Method (NDM) to the Solow growth financial model. The NDM is a numerical technique that combines the Natural Transform (NT) and the ADM-Adomian decomposition method. The NDM simplifies problem-solving by converting the original differential equations into algebraic equations as regards limitations associated with nonlinear models. From the results obtained by applying the NDM to the Solow growth financial model, researchers and policymakers can better understand the interplay between financial variables, such as savings rates, investment, and capital allocation, and their impact on economic growth dynamics, as a systematic approach to capturing the complex relationship between finance and economic development within the Solow framework is ensured. Further research and application of the NDM can contribute to advancing the knowledge of economic dynamics and support evidence-based decision-making in economic policy.

Stochastic Model Predictive Control Based on Polynomial Chaos Expansion With Application to Wind Energy Conversion Systems

The wind energy conversion system (WECS) has a complex structure, and its state space model is highly nonlinear. Due to the random uncertainty of wind speed, it poses a huge challenge to achieve optimal control tasks and ensure the safe and stable operation of the system. Therefore, this article proposes a stochastic model predictive control strategy based on Polynomial Chaotic Expansion (PCE), which achieves the control tasks of MPPT and constant power regions in wind energy conversion systems. Firstly, a simple algorithm is proposed to obtain a set of basis functions that are suitable for the stochastic variable wind speed. Then, the obtained basis functions are used to propagate the uncertainty of the original uncertain differential equation of the wind energy conversion system through polynomial chaotic expansion. Combining the operating region and constraint conditions of the wind energy conversion system, the original stochastic uncertainty problem is transformed into a deterministic convex optimization problem. Using NREL 5MW wind turbine as the research object for simulation, the task of capturing maximum wind energy in MPPT area and tracking rated power points in constant power area was achieved. The experimental results show that the proposed control method can effectively improve the wind energy capture capability and achieve accurate tracking of output power to rated power.

On the Temporal Tweezing of Cavity Solitons

Motivated by the work of Jang et al., Nat Commun 6:7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing.

Magneto-Hydrodynamic Stagnation Point Flow of Casson Williamson Hybrid Nanofluid Incorporating Viscous Dissipation and Suction/Injection Effect Past an Exponentially Stretching Cylinder

The ongoing inquiry aims to analyze stagnation point flow characteristics of magneto-hydrodynamic (MHD) Cason-Williamson hybrid nanofluids over an exponentially stretched cylinder, incorporating phenomena like viscous dissipation and suction/injection effects, as no prior investigation has been conducted on it, which represents the distinctiveness of the flow model. To facilitate analysis, the original partial differential equation (PDE) formulation of the flow model is transformed into non-dimensional ordinary differential equations (ODEs) employing dimensionless quantities, a process facilitated by the MATLAB bvp4c approach. Various non-dimensional variables are examined for their impacts on velocity profiles, temperature distribution, shearing stress, and Nusselt number. Results conveyed through graphs and detailed tables show thermal profile enhancement with escalating Weissenberg, Eckert, and Biot numbers for Casson Williamson hybrid nanofluid. Increasing copper nanoparticle volume in this fluid raises friction drag compared to the Casson hybrid nanofluid, with a 9% enhancement in shear stress. Conversely, the heat transport rate is reduced by about 1.5% for Casson Williamson hybrid nanofluid compared to Casson hybrid nanofluid. These findings significantly advance fluid dynamics and nanofluid exploration, offering opportunities for improved heat and mass transmission in various industries.

Averaging principle for Hifer–Katugampola fractional stochastic differential equations

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results.

Calculation of Plates of Different Shapes Using Generalized Equations of the Finite Difference Method

Statement of the problem. The article reviewed nonaxisymmetric tasks for calculating slabs in shape of a circle sector, wedge and a slab formed by a connection of rectangular and circular areas. The solution is based on generalized equations of the finite differences method. The algorithm allows us to take into account finite discontinuities of the desired function, the right side of the original differential equations, as well as discontinuities of derivatives of these functions without accumulation of the grid and using points outside the contour. It resolves oneself into a system of differential algebraic equations generated for field and contour points. In this paper the initial differential equations and their numerical approximation are provided. Results. Calculations of a semicircle and a wedge under different boundary conditions are presented, as well as an uncut slab in the form of a semicircle with an annular support and a composite slab are consi-dered. The solution of the above problems has been carried out on the minimum possible computational grid, comparison with the available solutions has been provided, and static checks of the calculation have been carried out. Conclusions. Along with the most common numerical method for calculating buildings and structures, it is possible and quite effective to use these equations. It is proved by means of some examples that such an approximation has a number of advantages: the results allow us to talk about the sufficient accuracy of the solution with a small number of partitions; fundamental simplicity and universality of the method.

Numerical-analytical approach to solving problems of non-stationary thermal conductivity of a non-thin annular plate

This paper considers the first stage of calculating the initial boundary value problem of non-stationary thermal conductivity of cylindrical bodies using a modified method of lines, namely dimension reduction of the original differential equations, initial and boundary conditions. The original equations of thermal conductivity are defined in a cylindrical coordinate system in a spatial setting. An object is a cylindrical body with commensurate dimensions. This area of research is relevant, because when calculating the load bearing elements of structures to thermal effects, the first step is to determine the distribution of temperature fields. Boundary conditions are considered as conditions of convective heat transfer, which by means of boundary transition are transformed into boundary conditions of the first and second types. Dimension reduction with respect to spatial coordinate is performed by the Bubnov-Galorkin-Petrov projection method using local basis functions. These functions are called "cover" functions, which are related to the lines drawn on the domain of the task. Normalized trigonometric series are used to reduce the dimensionality of the equations with respect to circular coordinate. All transformations are performed in index form. In addition to differential equations, the projection method reduces the dimensionality of the initial and boundary conditions. In this paper, the most optimal form of writing reduced equations is determined, which provides the ease of reducing the dimensionality of the original differential equations. Initial and boundary conditions take into account the impact of the environment. All this makes it possible to set a reduced initial boundary value problem for further calculation by numerical finite difference methods.

On Generalised Hankel Functions and a Bifurcation of Their Asymptotic Expansion

The generalised Bessel differential equation has an extra parameter relative to the original Bessel equation and its asymptotic solutions are the generalised Hankel functions of two kinds distinct from the original Hankel functions. The generalised Bessel differential equation of order ν and degree μ reduces to the original Bessel differential equation of order ν for zero degree, μ = 0. In both cases the differential equations have a regular singularity near the origin and the the point at infinity is the other singularity. The point at infinity is an irregular singularity of different degree, namely one for the original and two for the generalised Bessel differential equation. It follows that in the limit of degree being equal to zero the generalised Hankel functions do not converge to the original ones. The implication is that the generalised Bessel differential equation has a Hopf-type bifurcation for the asymptotic solution. In the case of a real variable and parameters the asymptotic solution is: (i) oscillatory when the degree of generalised Hankel function is zero (corresponding in this case to original Hankel functions); (ii) diverging hence unstable for the generalised Hankel functions with positive degree; (iii) decaying hence stable for the generalised Hankel functions with negative degree.

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor ($RL$), Resistor-Capacitor ($RC$), Resonant (Inductor-Capacitor) ($LC$), and Resistor-Inductor-Capacitor ($RLC$) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models.

Analytical solutions to the compressible Euler equations with cylindrical symmetry and free boundary

In this paper, we study the analytical solutions to the compressible Euler equations with cylindrical symmetry and free boundary. We assume that the free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. By using some ansatzs, we reduce the original partial differential equations into an ordinary differential equation about the free boundary. We prove that the free boundary grows linearly in time by constructing some new physical functionals. Furthermore, the analytical solutions to the compressible Euler equations with time-dependent damping are also considered and the spreading rate of the free boundary is investigated according to the various sizes of the damping coefficients.

Engine oil blended Casson hybrid nanofluid flow along a uniformly heated curved surface with Arrhenius activation energy and suction: A computational study

The research studies the numerical analysis of an incompressible Casson hybrid nanofluid, composed of Fe3O4–Cu nanoparticles blended with engine oil, flowing over an exponentially stretched curved surface. It explores the effects of an aligned magnetic field, a uniform heat source, activation energy, and suction while maintaining a low Prandtl number. By employing appropriate similarity transformations, the original partial differential equations are converted into ordinary ones and solved using bvp4c in MATLAB. Graphs and tables depict the influence of dimensionless factors on skin friction coefficient, velocity, thermal, and concentration profiles, as well as Nusselt and Sherwood numbers. The computational findings reveal that magnetic field alignment significantly modifies velocity, decreasing it while uniformly enhancing temperature and concentration profiles. Casson fluid behavior in engine oil amplifies temperature and concentration profiles, thereby enhancing heat and mass transfer rates. Moreover, an increased heat source elevates temperature but reduces overall heat transfer. The chemical reaction parameter diminishes concentration, whereas activation energy enhances it, albeit impacting the mass transfer rate conversely. Additionally, suction increases flow resistance, while the stretching parameter exhibits an inverse effect, both positively influencing heat and mass transfer rates at higher values. Practically, these findings offer insights crucial for optimizing heat exchangers, biomedical devices, and cooling systems, enhancing their efficiency and performance. Additionally, they inform the design of industrial processes involving fluid flow, aiding in the development of more effective and sustainable solutions.

Entropy optimization on magnetized radiative bioconvective flow of Carreau-Yasuda hybrid CNTs through a spherical surface in a non-Darcian porous medium with activation energy

This work inspects entropy generation and heat transfer induced by a bioconvection slip flow of nonlinear radiative Carreau-Yasuda hybrid nanofluid (NF) over a convectively heated sphere. Activation energy for microbes is contemplated in order to comprehend its contribution toward flow features. The original partial differential equations (PDEs) are rendered non-dimensional through appropriate conversions, and the resulting PDEs are solved using the overlapping grid spectral collocation algorithm. The impact of diverse flow factors on flow profiles, entropy generation, skin friction, heat, mass, and motile microbes transport rates is analyzed. Key outcomes reveal that hybrid NF flow demonstrates a supplementary indispensable feature in the operation of heat transport compared to mono NF flow. More entropy is generated in the system by using the hybrid NF model along with magnetic field, heat source, convective heating, nonlinear radiation, and viscous dissipation. Thermal fields and rate of heat transport are improved by including nonlinear radiative heat flux in the system. Mass and motile microbes transport rates are respectively enhanced by chemical and microbial reactions, but both quantities are reduced by activation energy. The effects of microbial reactions on flow quantities substantiate the significance of these features for the dynamics of microorganisms. The findings of this study can be useful in the upsurge of thermal performance of the working fluid and contribute toward the improvement of microbial fuel cell performance.

ПРОБЛЕМИ ТА ПЕРСПЕКТИВИ РОЗВИТКУ АВТОМАТИЗОВАНИХ СИСТЕМ УПРАВЛІННЯ ЦІЛЕВКАЗАННЯ

The justification of the need to improve the automated control process in the system of automatic targeting testing is presented. The modern Armed Forces of Ukraine use means of automation at all levels of command of the troops and managing the means of defeat. In parallel with the automation of the management process at all levels, the shadows of recent years indicate a shift of attention towards the lower management levels. The command systems at these levels collect, process and analyse information necessary to optimize the management of the actual combat assets. The specific fire task to combat crew, its transfer and delivery to the contractor is an important point in the process of such control. A central part of the task statement is targeting i.e. a brief, clear and understandable message about the location of the target and other information about it. The guaranteed transmission of target instruction from any level of control to the means of destruction is one of the directions of development and improvement of automated control systems for military purposes. The automated method for achieving the target-setting provides a minimum time for setting the firing task and carrying it out. However, in order for it to be revived, information on the full identification of targets at the different levels of control and sufficiently accurate information on the location of its forces and assets is necessary. In many cases, the dynamic properties of such control systems are described by high-order differential equations. The implementation of optimal speed in such cases faces a number of technical difficulties related to the need to calculate and implement the number of intervals of control action.The number of intervals is equal to the order of the differential equation of the system. In this case, it is advisable to move to a quasi-optimal control with the number of controls switching equal to three. Reducing the number of switches will not only simplify the functions of the microprocessor in such a control system, but also increase its reliability. Optimum system theory illuminates the way forward and points to unused reserves of existing automatic control systems. The development of mathematical theory is largely determined by the development of methods of synthesis of optimal systems. Bellman R. and Feldbaum O. first formulated and proved the theorem of n intervals or n switching by control systems. According to the n interval theorem, the number of switches can be reduced by replacing the n-order control system with an equivalent lower-order system. According to the hypothesis of Academician A.Y. Ishlinsky, a third-order system can be found for any n-order system. The paper presents a numerical experiment confirming the hypothesis of Academician A.Y. Ishlinsky about the possibility of presenting a high-order system equivalent to a third-order system. The problem of identifying the dynamics of technological processes is solved by acceleration curves. The use of direct numerical methods is limited in some cases by the inability to implement the input of the required form. In this case, object identification is carried out by a conventional differential equation of order 4. A type of differential equation of order 3 is also known. The values of the parameters at which the solutions of the differential equations of order 4 and order 3 are most closely related are found. To solve the problem, the Nelder-Mead method is used. The solution of the original differential equation is proposed to be found by the Runge-Kutt-Merson method. The obtained results make it possible to simplify the process of object management in the systems of automatic targeting testing. The numerical experiments carried out confirmed the conjecture of Academician A.Y. Ishlinsky that the n-order system can be represented by an equivalent system of order 3. Thus, to reduce the time required to collect, process and analyze information necessary for the management of combat assets.

Дополнительные граничные условия в задачах теплопроводности с переменным по координате начальным условием

It is exceedingly difficult to obtain mathematically accurate analytical solutions of heat conduction problems with a variable initial condition. Known solutions of these problems are expressed by cumbersome functional series that converge poorly in the range of small values of time and space variables. Thus, to obtain simpler and more effective solutions of these problems is an urgent issue. The authors have used an additional required function and additional boundary conditions to obtain solutions of the problem. Application of the additional required function allows us to reduce the original partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are in such a form that their fulfillment using the resulting solution is equivalent to the fulfillment of the equation at the boundary points. The authors have developed a technique to obtain an analytical solution of the heat conduction problem under a linear change of the initial condition, based on an additional required function and additional boundary conditions. Solution of an ordinary differential equation with respect to the additional required function determines the eigenvalues. In classical methods these eigenvalues are found in the solution of the Sturm–Liouville boundary value problem. The authors have proposed another, simpler solution to determine eigenvalues. An accurate analytical solution of the heat conduction problem for an unbounded plate with a coordinate-variable initial condition is obtained. The scientific and practical value of the proposed analytical solution is the development of an innovative approach to determine eigenvalues, as well as elimination of complex integrals when we solve the equation and initial conditions of the boundary value problem. It makes possible to simplify the use of the solution obtained in engineering applications.

Short delay limit of the delayed Duffing oscillator.

The delayed Duffing equation, x^{″}+ɛx^{'}+x+x^{3}+cx(t-τ)=0, admits a Hopf bifurcation which becomes singular in the limit ɛ→0 and τ=O(ɛ)→0. To resolve this singularity, we develop an asymptotic theory where x(t-τ) is Taylor expanded in powers of τ. We derive a minimal system of ordinary differential equationsthat captures the Hopf bifurcation branch of the original delay differential equation. An unexpected result of our analysis is the necessity of expanding x(t-τ) up to third order rather than first order. Our work is motivated by laser stability problems exhibiting the same bifurcation problem as the delayed Duffing oscillator [Kovalev etal., Phys. Rev. E 103, 042206 (2021)2470-004510.1103/PhysRevE.103.042206]. Here we substantiate our theory based on the short delay limit by showing the overlap (matching) between our solution and two different asymptotic solutions derived for arbitrary fixed delays.

Magneto-convective hybrid nanofluid slip flow over a moving inclined thin needle in a Darcy-Forchheimer porous medium with viscous dissipation

PurposeThe purpose of this study is to provide that porous media and viscous dissipation are crucial considerations when working with hybrid nanofluids in various applications.Recent years have witnessed significant progress in optimizing these fluids for enhanced heat transfer within porous (Darcy–Forchheimer) structures, offering promising solutions for various industries seeking improved thermalmanagement and energy efficiency.Design/methodology/approachThe first step is to transform the original partial differential equations into a system of first-order ordinary differential equations (ODEs). The fourth-order Runge–Kutta method is chosen for its accuracy in solving ODEs. The present study investigates the free convective boundary layer flow of hybrid nanofluids over a moving thin inclined needle with the slip flow brought about by inclined Lorentz force and Darcy–Forchheimer porous matrix, viscous dissipation.FindingsIt is found that slip conditions (velocity and Thermal) exist for a range of the natural convection boundary layer flow. In the hybrid nanofluid flow, which consists of Al2O3 and Fe3O4 are nanoparticles, H2O − C2H6O2 (50:50) are considered as the base fluid. The consequence of the governing parameter on the momentum and temperature profile distribution is graphically depicted. The range of the variables is 1 ≤ M ≤ 4, 1 ≤ d ≤ 2.5, 1 ≤ δ ≤ 4, 1 ≤ Fr ≤ 7, 1 ≤ Kr ≤ 7 and 0.5≤λ ≤ 3.5. The Nusselt number and skin friction factors are used to calculate the numerical values of various parameters, which are displayed in Table 4. These analyses elucidate that upsurges in the value of the Fr noticeably diminish the momentum and temperature. It is investigated to see if the contemporary results are in outstanding promise with the outcomes reported in earlier works.Practical implicationsThe results can be very helpful to improve the energy efficiency of thermal systems.Social implicationsThe hybrid nanofluids in heat transfer have the potential to improve the energy efficiency and performance of a wide range of systems.Originality/valueThis study proposes that in the combined effects of hybrid nanofluid properties, the inclined Lorentz force, the Darcy–Forchheimer model for porous media and viscous dissipation on the boundary layer flow of a conducting fluid over a moving thin inclined needle. Assessing the potential practical applications of the hybrid nanofluids in inclined needles, this could involve areas such as biomedical engineering, drug delivery systems or microfluidic devices. In future should explore the benefits and limitations of using hybrid nanofluids in these applications.

Дополнительные граничные условия в задачах теплопроводности с неоднородными граничными условиями

Using additional boundary conditions (ABC) and an additional sought function (ASF), a solution to the heat conduction problem with non-homogeneous boundary conditions has been obtained. ABC allows satisfying the differential equation at the boundaries, leading to its fulfillment both inside the domain and without resorting to integration over Cartesian coordinates. ASF reduces the original partial differential equation to a temporal ordinary equation, from which the eigenvalues of the boundary value problem are determined, as formulated in the classical methods of the Sturm-Liouville problem, stated in spatial variables. Thus, this work considers an alternative way of determining eigenvalues. The integration constants are found from the initial condition using the least squares method, which allows their values to be determined with a given accuracy. The solution obtained based on ABC and ASF approximates n → ∞ the exact analytical solution in the form of an infinite series, including trigonometric coordinate functions with coefficients that stabilize exponentially in time. In this case, the eigenvalues determined from the solution of the temporal ordinary differential equation regarding the additional sought function coincide with their exact values at any approximation. The accuracy of the integration constants, determined by the method of least squares, is controlled by the number of approximation points in the range of the auxiliary variable's variation. It should be noted that the additional boundary conditions considered in this work hold true for any other method of obtaining a solution to the problem under consideration, including the exact method, as can be verified by direct substitution. Therefore, their introduction does not distort the original mathematical formulation of the problem but only significantly simplifies the process of obtaining its analytical solution.

Определение параметров упругодемпфирующих устройств в механизмах карьерных экскаваторов

Introduction. The aim of the work is to develop a procedure for synthesizing the parameters of elastic-damping devices for shovel based on the representation of these devices in the form of additional feedbacks and optimization of the transfer functions of these feedbacks, which allows abstracting from specific methods of their implementation and making a comparative assessment and reasonable choice of device parameters. Research method. Optimization of the parameters of elastic-damping devices is carried out on the basis of using the root method for studying the characteristic polynomial of the transfer function of the additional feedback and setting the desired nature of the transient process, which is the solution of the original differential equation of motion with given roots. Results. A procedure for synthesizing the parameters of elastic-damping devices based on the representation of these devices in the form of additional feedbacks and optimization of the transfer functions of these feedbacks using root methods is proposed, which allows one to abstract from specific methods of their implementation and make a comparative assessment and reasonable choice of device parameters. Analytical dependencies are obtained, that establish a relationship between the parameters of elastic-damping devices and the quality of transient processes, making it possible to determine the effective range of these parameters. Numerical modeling of the electromechanical system of the digging mechanism of a mining excavator was carried out based on the use of its real characteristics, which confirmed the high efficiency of the proposed method for the synthesis of stiffness and damping parameters of elastic-damping devices. Currently, an experimental unit equipped with a prototype of an elastic-damping device is being created, on which it is expected to evaluate the performance and effectiveness of the proposed method for synthesizing parameters. After this, it is planned to contact potential consumers of such devices to determine the possibilities of practical application of both the devices themselves and the proposed method for synthesizing their parameters.

Entropy structure informed learning for solving inverse problems of differential equations

Entropy, since its first discovery by Ludwig Boltzmann in 1877, has been widely applied in diverse disciplines, including thermodynamics, continuum mechanics, mathematical analysis, machine learning, etc. In this paper, we propose a new method for solving the inverse XDE (ODE, PDE, SDE) problems by utilizing the entropy balance equation instead of the original differential equations. This distinguishing feature constitutes a major difference between our current method and other previous classical methods (e.g. SINDy). Despite concerns about the potential information loss during the compression procedure from the original XDEs to single entropy balance equation, various examples from MM reactions, Schlögl model and chemical Lorenz equations in the form of ODEs to nonlinear porous medium equation and Fokker–Planck equation with a double-well potential in the PDE form all well confirm the accuracy, robustness and reliability of our method, as well as its comparable performance with respect to SINDy.