Integration Of Differential Equations Research Articles

The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods.

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)≅Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

Vortex-Induced Nonlinear Bending Vibrations of Suspension Bridges with Static Wind Loads

A low stiffness makes long-span suspension bridges sensitive to loads, and this sensitivity is particularly significant for wind-induced nonlinear vibrations. In the present paper, nonlinear vibrations of suspension bridges under the combined effects of static and vortex-induced loads are explored using the nonlinear partial differential–integral equation that models the plane bending motion of suspension bridges. First, we discretized the differential–integral equation through the Galerkin method to obtain the nonlinear ordinary differential equation that describes the vortex-induced vibrations of the bridges at the first-order symmetric bending mode. Then, the approximate analytical solution of the ordinary differential equation was obtained using the multiple scales method. Finally, the analytical solution was applied to reveal the relationships between the vibration amplitude and other parameters, such as the static wind load, the frequency of dynamic load, structural stiffness, and damping. The results show that the static wind load slightly impacts the bridge’s vibrations if its influence on the natural frequency of bridges is ignored. However, the bridge’s vibrations are sensitive to the load frequency, structural stiffness, and damping. The vibration amplitude, as a result, may dramatically increase if the three parameters decrease.

An Improved Mathematical Theory for Designing Membrane Deflection-Based Rain Gauges

This paper is devoted to developing a more refined mathematical theory for designing the previously proposed membrane deflection-based rain gauges. The differential-integral equations governing the large deflection behavior of the membrane are improved by modifying the geometric equations, and more accurate power-series solutions of the large deflection problem are provided, resulting in a new and more refined mathematical theory for designing such rain gauges. Examples are presented to illustrate how to analyze the convergence of the power-series solutions and how to numerically calibrate membrane deflection-based linear rain gauges. In addition, some important issues are demonstrated, analyzed, and discussed, such as the superiority of the new mathematical theory over the old one, the reason why the classical geometric equations cause errors, and the influence of changing design parameters on the input–output relationships of rain gauges.

Multi-harmonic resonance of pipes conveying fluid with pulsating flow

The parametric excitation analysis of pipes conveying pulsating fluid usually focuses on the principal parametric resonance. In this paper, the principal parametric resonance and the combination resonance under parametric excitation are compared for the first time. A dynamic model of a fluid-conveying pipe with elastic torsional constraints at both ends is established. The nonlinear partial differential-integral equation is derived for governing the transverse vibration. The natural frequencies and modes of pipe are obtained. Moreover, the analytical solutions of stable region and nonlinear amplitude-frequency responses for three kinds of parametric resonances are derived. Then, the analytical results are verified numerically. The results show that there is no differential-type combination parametric resonance for pipes supported at both ends. Increasing material damping or torsional stiffness and decreasing mean velocity can increase the stable region of the pipe. However, it is interesting to note that when combination parametric resonance occurs between symmetric and antisymmetric modes, the effect of mean velocity on stability is reduced. In addition, more than one form of resonance occurs in some regions. The sum-type combination resonance is easier to be excited than the principal parametric resonance. Among them, the combination of first-fourth mode is the most sensitive to the torsional boundary, and its response is the largest. In summary, this paper demonstrates that the combination parametric resonance, which has not been researched deeply yet, can be more destructive than the principal parametric resonance.

Exact Solutions and Conservation Laws of A (2+1)-dimensional Combined Potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili Equation

This article investigates a sixth order integrable nonlinear partial differential equation model that fulfills the Hirota N-soliton. Space and time-dependent shift, rotation and space-dependent shift, time and space translations, and time and space dilations Lie point symmetries are presented methodically. Under a specific point symmetries, the Lie point symmetries lead to group invariant solutions. The significance of conservation laws of the underlying equation are shown. The results are quite accurate in recreating complex waves and the dynamics of their interactions.

Solving Some Integral and Fractional Differential Equations via Neutrosophic Pentagonal Metric Space

In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result.

On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J,S)-fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

Physics-Informed Neural Networks (PINNs) are artificial neural networks that encode Partial Differential Equations (PDEs) as an integral component of the ML model. PINNs are successfully used nowadays to solve PDEs, fractional equations, and integral–differential equations, including direct and inverse problems. Just as in the case of other kinds of artificial neural networks, the architecture, including the number and sizes of layers, activation functions, and other hyperparameters can significantly influence the network performance. Despite the serious work in this field, there are still no clear directions on how to choose an optimal network architecture in a consistent manner. In practice, expertise is required, with a significant number of manual trial and error cycles. In this paper, we propose PINN/GA (PINN/Genetic Algorithm), a fully automatic design of a PINN by an evolutionary strategy with specially tailored operators of selection, crossover, and mutation, adapted for deep neural network architecture and hyperparameter search. The PINN/GA strategy starts from the population of simple PINNs, adding new layers only if it brings clear accuracy benefits, keeping PINNs in the population as simple as possible. Since the examination of dozens of neural networks through the evolutionary process implies enormous computational costs, it employs a scalable computational design based on containers and Kubernetes batching orchestration. To demonstrate the potential of the proposed approach, we chose two non-trivial direct problems. The first is 1D Stefan transient model with time-dependent Dirichlet boundary conditions, describing the melting process, and the second is the Helmholtz wave equation over a 2D square domain. The authors found that PINNs accuracy gradually improves throughout the evolutionary process, exhibiting better performance and stability than parallel random search and Hyperopt Tree of Parzen Estimators, while keeping the network design reasonably simple.

Non-homogeneous impulsive time fractional heat conduction equation

This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

Bernoulli wavelet method for numerical solution of linear system of Fredholm integral equation of the second kind

One of the key tools for many fields of applied mathematics is the integral equations. Integral equations are widely utilized in many models, atmosphere–ocean dynamics, fluid mechanics, mathematical physics, and many other physical and engineering disciplines. A new numerical strategy based on the Bernoulli wavelet is introduced to solve system of Fredholm integral equations of second kind. In this paper, the Bernoulli wavelets are first built. Second, the system of Fredholm integral equations has been reduced into an algebraic system. In order to demonstrate the viability, and accuracy of the suggested Bernoulli wavelet approach, some numerical examples are offered at the end. The derived numerical results are examined with those from other numerical techniques and with exact solutions, demonstrating the superiority of the proposed method over those techniques. The novelty of proposed technique is that it can be extended for the numerical solution of two dimensional integral equations and differential equations appearing in engineering models, however some modifications will be required.

The capillary pressure curves from upscaling interfacial and unsaturated flows in porous layers with vertical heterogeneity

We provide the capillary pressure curves pc(s¯) as a function of the effective saturation s¯ based on the theoretical framework of upscaling unsaturated flows in vertically heterogeneous porous layers proposed recently (Z. Zheng, Journal of Fluid Mechanics, 950, A17, 2022). Based on the assumption of vertical gravitational-capillary equilibrium, the saturation distribution and profile shape of the invading fluid can be obtained by solving a nonlinear integral-differential equation. The capillary pressure curves pc(s¯) can then be constructed by systematically varying the injection rate. Together with the relative permeability curves k¯rn(s¯) that are already obtained. One can now provide quick estimates on the overall behaviours of interfacial and unsaturated flows in vertically-heterogeneous porous layers.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Asymptotic behavior of the solution of the boundary value problem for a singularly perturbed system of the integro-differential equations

In this paper, we study the asymptotic behavior of solutions to the boundary value problem for singularly perturbed systems of integro-differential equations. The aim of the work is to obtain an analytical formula, an asymptotic estimate of the solution of a boundary value problem, and to determine the asymptotic behavior of the solution by a smaller parameter at the starting point. The boundary value problem given in the paper is reduced to a boundary value problem posed in a singularly perturbed integral-differential equation of mixed type with respect to a fast variable. The Cauchy function, boundary functions and Green’s function of a singularly perturbed homogeneous differential equation are obtained, and their asymptotic estimates are also determined. With the help of these constructed functions, an analytical formula and an asymptotic estimate of this solution of the boundary value problem are obtained. The asymptotic behavior of the solution with respect to a small parameter is determined and the order of growth of its derivatives at the left point of a given segment is shown. It is established that the solution of the boundary value problem under consideration has an initial jump of zero order at the initial point.

Reentry vehicle friction heat estimated by means of Boltzmann integral-differential equation of fluctuation-dissipation theory

We begin with Atmosphere Einstein Brownian motions toward Knudsen Gas Boltzmann Kinetic formulism of the fluctuation-dissipation theory and applied the formulism to space travelers reentry vehicles. Early experimental results are estimated on the worst situation upper bound when a reentry vehicle is frictionally passing through a dense atmosphere. In reality, the vehicle has spent more time in a dilute atmosphere space. Accordingly, the fluctuation and dissipation theorem, derived from the Boltzmann kinetic equation, predicted a much less friction heat than the final reentry stage.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Positive Periodic Solution for Neutral-Type Integral Differential Equation Arising in Epidemic Model

This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin’s continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral differential equation. Compared with previous results, the existence conditions in this paper are less restricted, thus extending the results of the existing literature. Finally, two examples are given to show the effectiveness and merits of the main results of this paper. Our results can be used to obtain the existence of a positive periodic solution to the corresponding non-neutral-type integral differential equation.

Interpolation synthesis of thecontrollers for a multi-motorelectric drive system containingan elastically linked element

Partial differential equations, integral, differential, or other equations describe multi-motor automatic electric drive systems containing elastic conveyor belts. Because of the elastic and distributive nature of the system parameters, the transfer function describing them is often a complex expression, containing not only the arguments as a linear system but also the inertial and transcendental components. This makes the precise control of tension and speed synchronously much more complicated than the centralized parameter system. A promising numerical solution based on the real interpolation method will simplify the procedure for synthesizing control loops while preserving the characteristic properties of objects with distributed parameters. The objective of the study is to propose a feasible solution for synthesizing the regulators based on the real interpolation method; it allows direct operation with the original transfer function containing the inertial and transcendental components. In this paper, we proposed an approach to synthesize the control system for objects with distributed parameters using the real interpolation method to reduce computational capacity and synthesis error while preserving the properties of this object class. Building an experimental model of the two-motor electric drive system containing an elastic conveyor to verify the effectiveness of the proposed algorithm. The simulation and experimental results indicate that the control system with the received regulators operating stably and meets the required quality criteria. It proves the efficiency of the synthesis algorithm based on the real interpolation method.

Dissipative quintessence and its cosmological implications

We consider a generalization of the quintessence type scalar field cosmological models, by adding a multiplicative dissipative term in the scalar field Lagrangian, which generally is represented in an exponential form. The generalized dissipative Klein-Gordon equation is obtained in a general covariant form in Riemann geometry, from the variational principle with the help of the Euler-Lagrange equations. The energy-momentum tensor of the dissipative scalar field is also obtained from the dissipative Lagrangian, and its properties are discussed in detail. Several applications of the general formalism are presented for the case of the cosmological Friedmann-Lema\^{\i}tre-Robertson-Walker metric. The generalized Friedmann equations in the presence of the dissipative scalar field are obtained for a specific form of dissipation, with the dissipation exponent represented as the time integral of the product of the Hubble function, and of a function describing the dissipative properties of the scalar field. For this case the Friedmann equations reduce to a system of differential-integral equations, which, by means of some appropriate transformation, can be represented in the redshift space as a first order dynamical system. Several cosmological models, corresponding to different choices of the dissipation function, and of the scalar field potential, are considered in detail. For the different values of the model parameters the evolution of the cosmological parameters (scale factor, Hubble function, deceleration parameter, the effective density and pressure of the scalar field, and the parameter of the dark energy equation of state, respectively) is considered in detail by using both analytical and numerical techniques. A comparison with the observational data for the Hubble function and with the predictions of the standard $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ paradigm is presented for each dissipative scalar field model. In the large time limit the model describes an accelerating universe, with the effective negative pressure induced by the dissipative effects associated with the scalar field. Accelerated expansion in the absence of the scalar field potential is also possible, with the kinetic term dominating the expansionary evolution. The dissipative scalar field models describe well the data, with the model free parameters obtained by a trial and error method. The obtained results show the dissipative scalar field model offers an effective dynamical possibility for replacing the cosmological constant and for explaining the recent cosmological observational data.

Statistical analysis for solution of non-linear integro-differential equation by using odinary and accerlated technique of Kamal-Adomian Decomposition

The integro-differential equations have a significant role in presenting the daily life phenomenon and researchers used different approaches for the solution of such problems. But it has been noticed that nonlinear integro-differential equations lack closed-form solutions. Consequently, approximate approaches are essential for determining approximate solutions to these models. The foremost goal of this research is to offer an innovative combined method by employing the Kamal transform and the Adomian decomposition method (ADM) for extracting analytical estimated, closed form, and numerical results of the non-linear differential-integral equations. The proposed method is named the Kamal Adomian Decomposition Method (KADM). To evaluate its efficiency and consistency the outcomes attained by the offered method are compared with the Laplace decomposition method which shows that our method is more efficient, that reveals the reliability of the presented method. The KADM will be used in the future to analyze new systems that emerge in many branches of science. The convergence analysis of the series of solutions is also presented. Furthermore, we provide some interesting non-trivial examples to show the validity of our main results.