Equations Of Type Research Articles

Wave-breaking and persistence properties in weighted $$L^p$$ spaces for a Camassa–Holm type equation with quadratic and cubic nonlinearities

Wave-breaking and persistence properties in weighted $$L^p$$ spaces for a Camassa–Holm type equation with quadratic and cubic nonlinearities

Estimation of Optimal Lock-Down and Vaccination Rate of a Stochastic SIR Model: A Mathematical Approach

This paper utilizes a stochastic Susceptible-Infected-recovered (SIR) model with a nonlinear incidence rate to estimate the optimal lock-down intensity and vaccination rate under the COVID-19 pandemic environment. We use a Feynman-type path integral control approach to determine a Fokker-Plank type equation of this system. Since we assume the availability of information on the COVID-19 pandemic is complete and perfect, we show the existence of a unique fixed point. A non-linear incidence rate is used because, it can be raised from saturation effects that if the proportion of infected agents is very high so that exposure to the pandemic is inevitable, then the transmission rate responds slower than linearity to the increase in the number of infections. The simulation study shows that with higher diffusion coefficients susceptible and recovery curves keep the downward trends while the infection curve becomes ergodic. Finally, we perform a data analysis using UK data at the beginning of 2021 and compare it with our theoretical results.

On a Schrödinger–Kirchhoff Type Equation Involving the Fractional p-Laplacian without the Ambrosetti–Rabinowitz Condition

On a Schrödinger–Kirchhoff Type Equation Involving the Fractional p-Laplacian without the Ambrosetti–Rabinowitz Condition

A fast numerical method for the conductivity of heterogeneous media with Dirichlet boundary conditions based on discrete sine–cosine transforms

The aim of this work is to develop a fast numerical method for solving conductivity problems in heterogeneous media subjected to Dirichlet boundary conditions. The method is based on a fixed-point iterative solution to an integral Lippmann–Schwinger type equation that is obtained by a Galerkin discretization of the cell problem using an approximation space spanned by sine series; the solution field is split between a known term verifying the Dirichlet boundary conditions and an unknown term described by sine series, which is null on the boundary by construction. With a suitable numerical integration scheme of the elementary integrals involved in the Galerkin formulation, based on discrete sine–cosine transforms, the method relies on the numerical complexity of fast Fourier transforms. The method is assessed in several problems including composite materials and fibrous networks.

A Conservative Finite Element Scheme for the Kirchhoff Equation

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

Synthesis Weibull Stochastic Differential Equation: Properties and application

The Weibull distribution is suitable for modeling stochastic phenomena. However, most of the existing stochastic differential equations (SDEs) of Weibull type found in the literature do not follow the Weibull distribution, making them unsuitable for modeling phenomena that follow a Weibull random variable. To address this issue, this paper proposes a new SDE known as the Synthesis Weibull Stochastic Differential Equation (SWSDE) using the synthesis method. Some properties and numerical studies were carried out. The usefulness of the model is also demonstrated using Malawian wind speed data. The results show that the proposed SDE is suitable for predicting the data. The model also demonstrates a mean-reverting property, whereby the equilibrium is the solution of a differential equation in simulation studies. Additionally, the solutions of the model are found to always follow the Weibull distribution.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Existence and uniqueness of the global conservative weak solutions for a cubic Camassa-Holm type equation

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v=2arctan⁡ux along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u0∈H1(R) was proved.

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

A pure Stokes approach for coupling fluid flow with porous media flow

Most numerical approaches for coupling fluid flow with porous media flow rely either on Stokes equations in the fluid part of the domain and Darcy’s law in the porous part, or on Brinkman’s equation. In both cases, difficulties arise at the boundary between the two parts because the equations used in the porous part are not of Stokes type. In this paper, an alternative to Darcy’s law is proposed for modeling flows in porous media. This alternative relies on equations of Stokes type where the permeability tensor is replaced by force and stress derivative tensors. Numerical procedures are presented to compute these tensors from simulations at pore scale. Simulations in domains containing both fluid and porous parts are finally conducted simply assuming continuity of velocity and pressure and hence without imposing any condition at the boundary between the two parts. Results show that the proposed method is accurate and hence a promising alternative to Darcy’s law for problems involving both fluid and porous parts.

On the complex version of the Cahn–Hilliard–Oono type equation for long interactions phase separation

Abstract This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation.

Stein–Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein–Weiss type inequalities with general kernels on the upper half space and study the extremal functions of the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of positive solutions of corresponding Euler–Lagrange system. As an application, we derive some Liouville type results for the Hartree type equations on the half space.

A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy.

P-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

p-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

Low-dimensional Cox-Ingersoll-Ross process

The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. To derive this equation, we utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of a certain reflected Brownian motion. The equation mentioned above turns out to contain a term characterized by the local time of the corresponding reflected Brownian motion. Additionally, we establish a new connection between low-dimensional CIR processes and reflected Ornstein-Uhlenbeck (ROU) processes, providing a new representation of Skorokhod reflection functions.

On the S-asymptotically ω-periodic mild solutions for multi-term time fractional measure integro-differential equations

This paper deals with a class of nonlocal problem for multi-term time-fractional measure integro-differential evolution equations of mixed type in Hilbert spaces. Firstly, we introduce the concept of S-asymptotically ω-periodic mild solution, on the premise of by utilizing -resolvent family and measure functional (Henstock–Lebesgue–Stieltjes integral). Secondly, combining regulated functions and fixed point theorem, the existence S-asymptotically ω-periodic mild solution for a classes of measure-driven differential equation is discussed. Finally, as samples of applications, we consider the nonlocal problem to a class of time-fractional measure partial differential equation with nonlocal condition at the end of this paper.

Some properties for the fifth‐order Camassa‐Holm type equation

In this paper, we study several problems of the fifth‐order Camassa‐Holm type (FOCHT) equation. The local well‐posedness and blow‐up scenario are established at first. Then we prove the global existence under some conditions and analyze the large‐time behavior of the support of momentum density. Finally, we discuss the persistence property in Sobolev space.

New Local Fractional Mohand–Adomian Decomposition Method for Solving Nonlinear Fractional Burger’s Type Equation

In this article, we address the challenge of solving the nonlinear fractional Burger’s KdV equation, time-fractional Burger’s equation, and the fractional modified Burger’s equation. This is achieved by employing the Caputo and conformable derivatives. To tackle these equations, we introduce a new numerical method which is the combination of the local fractional Mohand transform and the Adomian decomposition method. This choice is driven by its straightforward methodology and reduced computational complexity. Moreover, to demonstrate the versatility of this technique, we provide several illustrative examples along with their corresponding exact or approximate solutions. These solutions are accompanied by graphical representations, further enhancing the clarity of the presented approach.

On the balanced pantograph equation of mixed type

UDC 517.9 We consider the balanced pantograph equation (BPE) y ′ ( x ) + y ( x ) = ∑ k = 1 m p k y ( a k x ) , where a k , p k > 0 and ∑ k = 1 m p k = 1. It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of <em>mixed type</em>, i.e., a 1 < 1 < a m , and prove that, in this case, the BPE has a nonconstant solution y and that y ( x ) ∼ c x σ as x → ∞ , where c > 0 and σ is the unique positive root of the characteristic equation P ( s ) = 1 - ∑ k = 1 m p k a k - s = 0. We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞ .

Research on dynamic response of centrifugal pendulum vibration absorber based on hybrid damping model

The application of the centrifugal pendulum vibration absorber (CPVA) has expanded from the aerospace sector to the automotive sector. To date, in most previous studies, viscous damping has been assumed to be present between the absorber and rotor, and damping has been neglected in other studies. To reflect and control the dynamic behaviour of the CPVA in vehicular applications realistically, a hybrid damping model incorporating viscous damping and rolling resistance was developed in this study and validated by conducting tests. Under the combined action of the centrifugal force, gravity, viscous resistance, and rolling resistance, an equation of motion of the CPVA was established using the Lagrangian function equation of the second type. The wear state of the kinematic pair between the absorber and rotor, which is common in practical applications, was included into a mathematical model in which the rolling resistance coefficient changes with the travel of the absorber, whereas the viscous resistance coefficient remains unchanged. A model was established to simulate the response of the absorber under a wide range of working conditions, and corresponding tests were performed. Compared with the results obtained using only viscous damping as reported in other studies, those of the proposed hybrid damping model are more consistent with the experimental results. This work fills the existing research gap and lays a foundation for further control of the dynamic behaviour of CPVAs in the gravitational field, particularly at low rotational speeds.