Third-order Partial Differential Equation Research Articles

Conservation laws and conserved quantities for the two-dimensional laminar wake

A systematic and unified method is presented for the derivation of the conserved quantities for the laminar classical wake and the wake of a self-propelled body. The multiplier method for the derivation of conservation laws is applied to the far downstream wake equations expressed in terms of the velocity components which gives rise to a second-order system of two partial differential equations, and in terms of the stream function which results in one third-order partial differential equation. Once the corresponding conservation laws are obtained, they are integrated across the wake and upon imposing the boundary conditions and the condition that the drag on a self-propelled body is zero, the conserved quantities for the classical wake and the wake of a self-propelled body are derived. In addition, this method results in the discovery of a new laminar wake which may have physical significance.

A Comparative Study of Numerical Methods for Solving (n+1) Dimensional and Third-Order Partial Differential Equations

A Comparative Study of Numerical Methods for Solving (<i>n</i>+1) Dimensional and Third-Order Partial Differential Equations

Index transforms with the squares of Bessel functions

ABSTRACTNew index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained.

The Third-Order Viscoelastic Acoustic Model Enables an Ice-Detection System for a Smart Deicing of Wind-Turbine Blade Shells

The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-relaxation time (SRT) for the material and is applicable at any value of the SRT. The notion of a smart deicing system (SDS) for blade shells (BSs) of a wind turbine is specified. The work considers the stress in a BS as the one caused by the operational load on the BS. The work develops key design issues of a prospective ice-detection system (IDS) able to supply an array of the heating elements of an SDS with the element-individual spatiotemporal data and procedures for identification of the material parameters of atmospheric-ice (AI) layer accreted on the outer surfaces of the BSs. Both the SDS and IDS flexibly allow for complex, curvilinear and space-time-varying shapes of BSs. The proposed IDS presumes monitoring of the QE components of the normal stresses in BSs. The IDS is supposed to include an array of pressure-sensing resistors, also known as force-sensing resistors (FSRs), and communication hardware, as well as the parameter-identification software package (PISP), which provides the identification on the basis of the aforementioned PDE and the data measured by the FSRs. The IDS does not have hardware components located outside the outer surfaces of, or implanted in, BSs. The FSR array and communication hardware are reliable, and both cost- and energy-efficient. The present work extends methods of structural-health/operational-load monitoring (SH/OL-M) with measurements of the operational-load-caused stress in closed solid shells and, if the prospective PISP is used, endows the methods with identification of material parameters of the shells. The identification algorithms that can underlie the PISP are computationally efficient and suitable for implementation in the real-time mode. The identification model and algorithms can deal with not only the single-layer systems such as the BS layer without the AI layer or two-layer systems but also multi-layer systems. The outcomes can be applied to not only BSs of wind turbines but also non-QE closed single- or multi-layer deformable solid shells of various engineering systems (e.g., the shells of driver or passenger compartments of ships, cars, busses, airplanes, and other vehicles). The proposed monitoring of the normal-stress QE component in the mentioned shells extends the methods of SH/OL-M. The topic for the nearest research is a better adjustment of the settings for the FSR-based measurement of the mentioned components and a calibration of the parameter-identification model and algorithms, as well as the resulting improvement of the PISP.

Structural changes of small amplitude kinetic Alfvén solitary waves due to second-order corrections

The structural changes of kinetic Alfvén solitary waves (KASWs) due to higher-order terms are investigated. While the first-order differential equation for KASWs provides the dispersion relation for kinetic Alfvén waves, the second-order differential equation describes the structural changes of the solitary waves due to higher-order nonlinearity. The reductive perturbation method is used to obtain the second-order and third-order partial differential equations; then, Kodama and Taniuti's technique [J. Phys. Soc. Jpn. 45, 298 (1978)] is applied in order to remove the secularities in the third-order differential equations and derive a linear second-order inhomogeneous differential equation. The solution to this new second-order equation indicates that, as the amplitude increases, the hump-type Korteweg-de Vries solution is concentrated more around the center position of the soliton and that dip-type structures form near the two edges of the soliton. This result has a close relationship with the interpretation of the complex KASW structures observed in space with satellites.

On the Einstein-Weyl and conformal self-duality equations

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as “master dispersionless systems” in four and three dimensions, respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note, we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar partial differential equations for a general anti-self-dual conformal structure in neutral signature.

The explicit probability distribution of the sum of two telegraph processes

We consider two independent Goldstein–Kac telegraph processes X1(t) and X2(t) on the real line ℝ, both developing with constant speed c &gt; 0, that, at the initial time instant t = 0, simultaneously start from the origin 0 ∈ ℝ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ &gt; 0. Closed-form expressions for the transition density φ(x, t) and the probability distribution function Φ(x, t) = Pr {S(t) &lt; x}, x ∈ ℝ, t &gt; 0, of the sum S(t) = X1(t) + X2(t) of these processes at arbitrary time instant t &gt; 0, are obtained. It is also proved that the shifted time derivative g(x, t) = (∂/∂t + 2λ)φ(x, t) satisfies the Goldstein–Kac telegraph equation with doubled parameters 2c and 2λ. From this fact it follows that φ(x, t) solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.

Solution of fractional third-order dispersive partial differential equations

In this paper, we proposed fractional differential transform method(FDTM) and modified fractional differential transform method(MFDTM) for the solution of fractional third-order dispersive partial differential equations in one- and higher-dimensional spaces. The plotted graphs illustrate the behavior of the solution for different values of fractional orderα. The efficiency and accurateness of the proposed methods are examined by means of four numerical experiments.

Mathematical Modelling of Surfactant Self-assembly at Interfaces

We present a mathematical model to describe the distribution of surfactant pairs in a multilayer structure beneath an adsorbed monolayer. A mesoscopic model comprising a set of ordinary differential equations that couple the rearrangement of surfactant within the multilayer to the surface adsorption kinetics is first derived. This model is then extended to the macroscopic scale by taking the continuum limit that exploits the typically large number of surfactant layers, which results in a novel third-order partial differential equation. The model is generalized to allow for the presence of two adsorbing boundaries, which results in an implicit free-boundary problem. The system predicts physically observed features in multilayer systems such as the initial formation of smaller lamellar structures and the typical number of layers that form in equilibrium.

Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws

This paper aims to construct conservation laws for a Benjamin–Bona–Mahony equation with variable coefficients, which is a third-order partial differential equation. This equation does not have a Lagrangian and so we transform it to a fourth-order partial differential equation, which has a Lagrangian. The Noether approach is then employed to construct the conservation laws. It so happens that the derived conserved quantities fail to satisfy the divergence criterion and so one needs to make adjustments to the derived conserved quantities in order to satisfy the divergence condition. The conservation laws are then expressed in the original variable. Finally, a conservation law is used to obtain exact solution of a special case of the Benjamin–Bona–Mahony equation.

Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

Direct numerical methods for solving a class of third-order partial differential equations

In this paper, three types of third-order partial differential equations (PDEs) are classified to be third-order PDE of type I, II and III. These classes of third-order PDEs usually occur in many subfields of physics and engineering, for example, PDE of type I occurs in the impulsive motion of a flat plate. An efficient numerical method is proposed for PDE of type I. The PDE of type I is converted to a system of third-order ordinary differential equations (ODEs) using the method of lines. The system of ODEs is then solved using direct Runge–Kutta which we derived purposely for solving special third-order ODEs of the form y‴=f(x,y). Simulation results showed that the proposed RKD-based method is more accurate than the existing finite difference method.

Series solutions of partial differential equations using the complex integral method

The complex integral method for solving ordinary differential equations in series [3, 7, 8] is extended to cover the series solution of partial differential equations also. The means of this extension is straightforward, with both ‘ordinary’ and ‘Frobenius’ multiple variable power series being dealt with. Standard examples of the application of the extended method(s) to first-order, second-order, third-order and simultaneous partial differential equations are provided throughout. Examples also include the series solution of a non-linear partial differential equation and the consideration of series solutions with negative powers.

Numerical Algorithm for the Third-Order Partial Differential Equation with Three-Point Boundary Value Problem

A numerical method based on the reproducing kernel theorem is presented for the numerical solution of a three-point boundary value problem with an integral condition. Using the reproducing property and the existence of orthogonal basis in a new reproducing kernel Hilbert space, we obtain a representation of exact solution in series form and its approximate solution by truncating the series. Moreover, the uniform convergency is proved and the effectiveness of the proposed method is illustrated with some examples.

Method of Lines for Third Order Partial Differential Equations

The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.

Nonlinear dynamics investigation in parameter planes of a periodically forced compound KdV-Burgers equation

Parameter plane plots related to a periodically forced compound Korteweg-de Vries-Burgers system, which is modeled by a third-order partial differential equation, are reported. It is shown that typical periodic structures embedded in a chaotic region in these parameter planes, organize themselves in different ways. There are bifurcation sequences whose periods have a well-defined law of formation, that may be written in a closed form, and there are bifurcation sequences self-organized in period-adding cascades.

Two anisotropic fourth‐order partial differential equations for image inpainting

In this study, the authors propose two fourth-order partial differential equations (PDEs) to inpaint the image. By analysing those anisotropic fourth-order PDEs and comparing their diffusion images, the authors confirm they are forward diffusion or backward diffusion. A numerical algorithm is presented using a finite-difference method and analyse the stability of discretisation. Finally, they show various experimental results and conclude that the proposed new models are better than the second-order and third-order PDEs, especially for weakening the blocky effects.

A novel technique for solving Cauchy problem for the third-order linear dispersive partial differential equation

To solve Cauchy problem for the third-order linear dispersive partial differential equation, the reduced differential transform (RDT) method is used. Without using any discretization or perturbation the technique gives analytical approximation in the form of rapidly convergent sequence with well-structured terms that can usually be expressed in a compact form. The efficiency and reliability of the method is demonstrated through a variety of homogeneous/ non-homogeneous, one-, two-, and three-dimensional model problems. Comparison of the results by RDT method and those obtained through variational iteration method (VIM) signifies that RDT method is more efficient and rapidly convergent. Moreover it is computationally an inexpensive method. Key words: Cauchy problem, dispersive partial differential equation, reduced differential transformation

Permeability Models for Magma Flow through the Earth's Mantle: A Lie Group Analysis

The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.

Gauss curvature-driven image inpainting for image reconstruction

In this paper, we propose a third-order Gauss curvature-driven geometric diffusion Partial Differential Equation for inpainting and reconstructing images. In Gauss curvature-driven diffusion processes, the rate of diffusion is directly proportional to the Gauss curvature value of the level curve. Since the Gauss curvature is the product of principal curvatures, its value become zero when even one of the principal curvatures is zero. Therefore, when Gauss curvature is used as a driving function for diffusion, the evolution preserves some of the meaningful structures with nonzero mean curvature values (viz. curvy edges, corners, etc.). However, the noise features always have nonzero Gauss curvature value and hence the diffusion process effectively removes them. The inpainting property of geometric PDE based on the Gauss curvature is being used in this work for reconstructing lost or degraded information. A filter is proposed to reconstruct the original images from the observed blurred and noisy images along with inpainting the desired image domain.