Original Differential Equation Research Articles

Nonparametric probabilistic forecasting based stochastic optimal scheduling of integrated electricity and gas systems

The volatility and sporadic availability of renewable energy create significant challenges to the optimal scheduling of integrated electricity and gas systems (IEGS). This paper develops a nonparametric probabilistic forecasting based stochastic scheduling approach of IEGS. The quantile at a series of quantile levels can be generated by direct quantile regression method. Given the set of predicted quantiles, a set of representative scenarios for wind power uncertainty can be obtained by using Monte Carlo simulation method and scenario reduction approach. Based on the implicit finite difference scheme, the original partial differential equations of the gas network are discretized to establish an algebraic model, which provide possibility for efficient solution. Then, the nonconvexity caused by the momentum equation is eliminated by the second-order core relaxation. Finally, the stochastic optimal scheduling model is reformulated as a second-order core programming problem. Numerical simulations are performed to showcase the superiority of the established stochastic optimal scheduling model.

Entropy optimization in mixed convective boundary layer flow of CNTs Casson nanoliquid through a vertical cylinder moving with nonlinear velocity

This article inspects entropy generation and mixed convection boundary layer flow of carbon nanotubes (CNTs) Casson nanoliquid via semi-infinite vertical cylinder which moves with nonlinear velocity in Darcy-Forchheimer porous medium. Appropriate similarity variables have been employed to convert the original partial differential equations into ordinary differential equations that have been solved using overlapping grid spectral quasilinearisation method (SQLM). Comparison of accuracy, convergence and stability between the two methods is made. Rate of entropy generated, flow fields and engineering quantities are discussed for different embedding parameters. It is revealed that single-wall CNTs are more efficacious to improve heat transport features than multi-wall CNTs. The fluid flow and thermal dispersion processes improve with consideration of curved surface, CNTs, non-Newtonian fluid and injection of the fluid. The curvature surface and suction of the fluid contribute towards growth of skin friction factor and rate of thermal transference. Entropy generated expands by accounting for viscous nature of the fluid, strong nonlinearity, suction, non-Newtonian fluid, convective boundary condition and CNTs. The study finds applications in various processes, which are massively impacted by heat transport enhancement and high porosity. Boundary layer flow and heat transfer analysis through cylinders are relevant to various metallurgical and engineering solicitations.

Lie analysis and nonlinear propagating waves of the (3 + 1)-dimensional generalized Boiti–Leon–Manna–Pempinelli equation

The (3+1)-dimensional generalized Boiti-Leon-Manna-Pempinelli equation, which describes the evolution of Riemann wave propagation in an in-compressible fluid along three mutually perpendicular axes, is investigated in this research article. The study encompasses the analysis of Lie symmetries, corresponding symmetry reductions, and conservation laws associated with the equation. The Lie symmetry method is employed to determine the infinitesimal generators of the considered equation. Furthermore, commutator table is generated, and symmetry groups corresponding to each infinitesimal generator are derived. By utilizing the similarity reduction technique, the original nonlinear partial differential equation is transformed into nonlinear ordinary differential equations. Then, generalized exp(−ϕ(ζ)) expansion technique is utilized to solve the reduced equations and estimate specific traveling wave solutions for the equation. Moreover, graphical representations are employed to illustrate the traveling wave solutions, employing suitable parameter values. Additionally, the multiplier approach is utilized to calculate conserved vectors for the equation under consideration.

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

A Second-Order Difference Scheme for Solving a Class of Fractional Differential Equations

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

Solution of filtration problems of construction geotechnology in thin water layers

Filtration problems of construction geotechnics during water movement in thin permeable layers of soils and artificial materials are considered. Under such conditions, traditional approaches borrowed from classical hydrogeology give errors. This is due to the fact that under normal conditions of underground aquifers of a sufficiently large depth, in hydrogeological practice, the hydraulic theory of filtration is used, in which the velocities of filtration currents are averaged along the vertical coordinate. The original nonlinear differential equations of filtration flows are subjected to forced linearization by one or another known method. In this case, the method of linearization by introducing the average power (depth) of the filtration flow is widely used. This method was originally developed for aquifers of great depth. In the presented work, an attempt was made to abandon such a linearization and solve some of the problems of unsteady filtration flows in thin aquifers, which are characteristic of building geotechnics, with obtaining analytical solutions of the final form without introducing artificial simplifications. A methodology is proposed for combining analytical and numerical methods in the process of finding solutions to filtration problems of construction geotechnics with obtaining self-similar dependencies. This approach will make it possible to avoid linearization errors and carry out filtration calculations in thin permeable layers of soils and artificial materials more accurately. A review of domestic and foreign scientific publications on the topic of research has shown some stagnation in this direction. The comparison revealed that in thin permeable layers of soils and artificial materials, preference can be given to filtration calculations using the presented method, since it more accurately describes the non-stationary movement of water in construction geotechnics.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

An Alternative Method for Analytical Solutions of Two-Dimensional Black-Scholes-Merton Equation

We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.

Backstepping Control for Vibration Suppression of 2-D Euler–Bernoulli Beam Based on Nonlinear Saturation Compensator

In this article, a boundary control scheme is proposed to suppress 2-D vibration of Euler–Bernoulli beam with output constraints and input saturation. The original partial differential equations (PDEs) model is transformed to a new form containing virtual control. Then a boundary controller is designed via the backstepping method to suppress the coupled vibration. The hyperbolic tangent function and Nussbaum function are employed to deal with the input saturation. A barrier Lyapunov function with time adjusting function is introduced to suppress the structural vibration of Euler–Bernoulli beam with arbitrary initial conditions. The disturbance observer is designed to deal with the unknown boundary disturbance. Finally, the simulation results show the effectiveness of the proposed vibration controller.

Near-resonance approximation of rotating Navier–Stokes equations

We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any H s space with . Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus’s aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.

Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection

Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.

Exploring complicated behaviors of a delay differential equation

<abstract><p>Complicated behaviors of a delay differential equation are explored through the Euler discretization method. It rigorously shows that the corresponding discrete equation can be chaotic under some conditions, which reflects that there exist complicated behaviors in the original delay differential equation.</p></abstract>

Numerical Methods That Preserve a Lyapunov Function for Ordinary Differential Equations

The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e., numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is implemented for the numerical integration of a system of ordinary differential equations. In principle, this procedure yields first-order methods, but the analysis paves the way for the design of higher-order methods. As a case in point, the proposed method is applied to the Duffing equation without external forcing, considering that, in this case, preserving the Lyapunov function is more important than the accuracy of particular trajectories. Results are validated by means of numerical experiments, where the discrete gradient method is compared to standard Runge–Kutta methods. As predicted by the theory, discrete gradient methods preserve the Lyapunov function, whereas conventional methods fail to do so, since either periodic solutions appear or the energy does not decrease. Moreover, the discrete gradient method outperforms conventional schemes when these do preserve the Lyapunov function, in terms of computational cost; thus, the proposed method is promising.

Calculating a beam of variable section lying on an elastic foundation

In this article, there has been studied the bending state of a reinforced concrete beam with a variable cross-sectional height along its length that rests along its entire length on a brick wall. The beam is under the action of arbitrarily located concentrated forces. The study has been performed on the basis of the original inhomogeneous differential equation of the 4th order, taking into account the external load and the bedding coefficients of the elastic foundation. Using the finite difference method, typical resolving finite difference equations have been obtained. A study of the influence of the degree of elasticity of the base, with a change in the value of the bed coefficients and elasticity parameters, was conducted. The results confirm the reliability of theoretical and practical calculations. Given theoretical provisions and applied results can be used in scientific research in the field of mechanics of a deformable solid body, as well as in practical design.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

Sermaye Birikiminin Fonksiyonel İfadesi ve Solow Uzun-Dönem Büyüme Modelinde Kararlı-Denge Koşulları

This paper proposes a general solution to Solow’s original differential equation explaining the rate of change of capital-labor ratio. Determining the time path of capital-labor ratio, we obtain novel general conditions under which the capital-labor ratio can reach a stable steady-state value. Finally, we apply our findings to the Solow’s CES production function example and we demonstrate that in addition to identification of steady-state conditions, we can obtain exact time period when the economy can reach its stable steady-state capital per labor magnitude. Our results state some policy implications that can be outlined as follows. First, in economies where elasticity of substitution between capital and labor is lower than unity, the economic policies should be different than those implemented in economies where elasticity is greater than unity. Second, the main source of uncertainty in perspective of policy makers is not to determining exactly when the economy activity would reach a stable steady-state path and our findings aim to shed light on this uncertainty.

The Discussion About the Identification of the Anand Model Parameters and Two Alternative Identification Methods

Abstract The Anand model is a unified viscoplastic model, which is widely employed to describe the solder material deformation. The parameters in the Anand model for a certain material are usually identified by using the classical method based on two algebraic equations derived from the original differential equation of the Anand model. However, the second algebraic equation describing the relationship between the stress and inelastic strain is obtained with some terms about the unsteady value of internal variable neglected. But the effects induced by the omission of some unsteady terms on the effectiveness of classical method are not researched comprehensively. Therefore, in this paper, the effects of the omitted terms on the accuracy of the classical method are discussed. The inelastic deformation for the material which the second algebraic equation cannot describe due to the omission of unsteady terms is presented. The precondition for obtaining accurate results from the second algebraic equation is given out. Two criteria used to judge the effectiveness of the second algebraic equation are derived. To reduce the error related to the second algebraic equation of the classical method for some materials, two alternative identification methods are proposed. By combining the step of solving differential equation and genetic algorithm, the parameters in the Anand model originally identified by the second algebraic equation are determined in the processes of the two proposed methods. The effectiveness of the two alternative methods is presented by identifying the material Anand parameters where the classical method cannot be applied.

Two-parameter bifurcations analysis of a delayed high-temperature superconducting maglev model with guidance force.

A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf-Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf-Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model.

Fractional Differential Equations in the Exploration of Geological and Mineral Construction

Abstract As the geological exploration data is relatively sparse, unevenly distributed, and contains many geological faults, simple geological surface reconstruction has certain limitations. Based on the fractional differential equations, the paper establishes a subsidence prediction model in exploring geological and mineral resources. The dynamic system described by the reaction-diffusion equation can be mapped to a nonlinear cellular network through space and time discretization. At the same time, the original partial differential equations can be transformed into ordinary differential equations. Furthermore, we can use the difference method to simulate its evolutionary behavior quantitatively. The research results show that the error accuracy between the prediction results of the fractional gray theory established in this paper and the actual engineering results is higher.