Constant-coefficient Equation Research Articles

Error estimate of a fully decoupled numerical scheme based on the Scalar Auxiliary Variable (SAV) method for the Boussinesq system

In this paper, we will investigate the unconditional stability and error estimate of the fully decoupled numerical scheme for the Boussinesq equations. The newly constructed numerical scheme is based on the pressure correction technique and the SAV method, in which all coupling terms and nonlinear terms are completely decoupled, that is, we only need to solve several linear constant-coefficient equations. We rigorously prove the unconditional stability and convergence of the time-marching scheme and discuss all the details of the algorithm implementation. Finally, we implement some numerical experiments to verify its stability and accuracy.

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

In this paper, we propose two theorems to illustrate the types of equations that can be solved using the quadratic function method to derive the lump solutions localized in the whole plane, which are called three-wave lump solutions, and provide two constant-coefficient equations to illustrate. We further extend the quadratic function method to the variable-coefficient differential equations and obtain the three-wave lump solutions for two (3+1)-dimensional variable-coefficient equations. Moreover, the amplitudes of these lump waves and the distances between the two valleys of each lump are also obtained. Meanwhile, the motion trails, displacements and the velocities of these lump waves are analyzed in detail by virtue of numerical simulation. The study can be used to describe the motion of nonlinear waves in shallow water under the influence of time, and the results can enrich the types of solutions for the KdV-type equations. In addition, the 3d plots and corresponding density plots of the lump waves are displayed to show their spatial structures.

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

Series Solution Method for Solving Sequential Caputo Fractional Differential Equations

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order q,0<q<1, and this solution matches with the integer solution for q=1. In addition, we also developed a series solution method for a linear sequential Caputo fractional differential equation with constant coefficients of order 2q, which is sequential for order q with Caputo fractional initial conditions. The advantage of our method is that the fractional order q can be used as a parameter to enhance the mathematical model, compared with the integer model. The methods developed here, namely, the series solution method for solving Caputo fractional differential equations of constant coefficients, can be extended to Caputo sequential differential equation with variable coefficients, such as fractional Bessel’s equation with fractional initial conditions.

A NumericalInvestigation for a Class of Transient-State Variable Coefficient DCR Equations

In this paper, a combined Laplace transform (LT) and boundary element method (BEM) is used to find numerical solutions to problems of anisotropic functionally graded media that are governed by the transient diffusion–convection–reaction equation. First, the variable coefficient governing equation is reduced to a constant coefficient equation. Then, the Laplace-transformed constant coefficients equation is transformed into a boundary-only integral equation. Using a BEM, the numerical solutions in the frame of the Laplace transform may then be obtained from this integral equation. Then, the solutions are inversely transformed numerically back to the original time variable using the Stehfest formula. The numerical solutions are verified by showing their accuracy and steady state. For symmetric problems, the symmetry of solutions is also justified. Moreover, the effects of the anisotropy and inhomogeneity of the material on the solutions are also shown, to suggest that it is important to take the anisotropy and inhomogeneity into account when performing experimental studies.

The subdivision-based IGA-EIEQ numerical scheme for the binary surfactant Cahn–Hilliard phase-field model on complex curved surfaces

In this paper, we consider numerical approximations of the binary surfactant phase-field model on complex surfaces. Consisting of two nonlinearly coupled Cahn–Hilliard type equations, the system is solved by a fully discrete numerical scheme with the properties of linearity, decoupling, unconditional energy stability, and second-order time accuracy. The IGA approach based on Loop subdivision is used for the spatial discretizations, where the basis functions consist of the quartic box-splines corresponding to the hierarchic subdivided surface control meshes. The time discretization is based on the so-called explicit-IEQ method, which enables one to solve a few decoupled elliptic constant-coefficient equations at each time step. We then provide a detailed proof of unconditional energy stability along with implementation details, and successfully demonstrate the advantages of this hybrid strategy by implementing various numerical experiments on complex surfaces.

Traveling Waves in Shallow Seas of Variable Depths

The problem of the existence of traveling waves in inhomogeneous fluid is very important for enabling an explanation of long-distance wave propagations such as tsunamis and storm waves. The present paper discusses new solutions to the variable-coefficient wave equations describing traveling waves in fluid layers of variable depths (1D shallow-water theory). Such solutions are obtained by using the transformation methods when variable-coefficient equations can be reduced to the constant coefficient equation when the existence of traveling waves is evident. It is shown that there is a wide class of monotonic bottom profiles (discrete set) that allow the existence of traveling waves that are not reflected in a strongly inhomogeneous water medium. Their temporal shape changes with distance, mainly near the water–land boundary (shoreline). Traveling waves can transfer the wave energy over a long distance that is often observed at the transoceanic propagation of tsunami waves.

Novel, linear, decoupled and unconditionally energy stable numerical methods for the coupled Cahn–Hilliard equations

This paper uses a novel numerical approach to approximate the coupled Cahn–Hilliard equations, which are a highly nonlinear system depicting the phase separation of the homopolymer and copolymer mixtures. The new method is named 3S-IEQ, and its construction and calculation are more straightforward than the invariant energy quadratization and scalar auxiliary variable methods. Notably, we only need to solve two linear decoupled constant-coefficient equations at each time step. Numerical simulations are shown

Equivalence transformations of a generalized fifth-order KdV equation with variable coefficients

By using the infinitesimal method, we firstly obtain the equivalence transformations for a class of generalized fifth-order KdV (gfKdV) equation with variable coefficients. With the application of equivalence transformations, the number of variable coefficients is reduced and the most general form which can be mapped by an equivalence transformation to the constant-coefficient equation is explored. Furthermore, the transformational properties of the gauged subclass are proposed via the direct method. We also construct the transformations that connect the three integrable variable coefficient gfKdV equations obtained before by Painlev e ́ test to their integrable constant-coefficient counterparts.

Construction and analysis of series solutions for fractional quasi-Bessel equations

In this paper we introduce fractional quasi-Bessel equations $$\begin{aligned} \sum _{i=1}^{m}d_i x^{\xi _i}D^{\alpha _i} u(x) + (x^\beta - \nu ^2)u(x)=0 \end{aligned}$$and construct their existence theory in the class of fractional series solutions. In order to find the parameters of the series, we derive the characteristic equation, which is surprisingly independent of the terms with non-matching parameters \(\xi _i\ne \alpha _i\). Our methodology allows us to obtain new results for a broad class of fractional differential equations including quasi-Euler equations. As a particular example, we demonstrate how our approach works for the constant-coefficient equations. The theoretical results are justified computationally.

Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods

In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $\epsilon$ such that $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding $u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion $\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model

In this article, we construct a fully discrete finite element numerical scheme with linearity, decoupling, unconditional energy stability, and second-order time accuracy for the Navier-Stokes coupled phase-field crystal model. The key idea is based on the design of several auxiliary ODEs, combined with the finite element method for spatial discretization, the projection method for the Navier-Stokes equations, and the IEQ type method for the nonlinear potentials. At each time step, by using the nonlocal splitting technique, only a few decoupled elliptic constant-coefficient equations need to be solved. We further prove that the developed scheme is unconditionally energy stable, and the detailed implementation process is given as well. To verify the effectiveness of the developed scheme, various numerical experiments are carried out, including the crystal growth under the action of shear flow and the sedimentation process of a large number of particles.

A Generalized Method of Undetermined Coefficients

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples.

An LT-BEM formulation for problems of anisotropic functionally graded materials governed by transient diffusion–convection–reaction equation

Problems of anisotropic functionally graded media which are governed by the transient diffusion–convection–reaction equation of spatially varying coefficients are discussed in this paper. A mathematical analysis is used to transform the variable coefficient equation into a constant coefficient equation. A boundary-only integral equation is then derived from this constant coefficient equation after being Laplace transformed. Numerical solutions to the problems are sought by using a boundary element method (BEM) which is combined with the Stehfest formula for the numerical Laplace transform inversion. Some problems considered are those of compressible or incompressible flow, and of media which are quadratically, exponentially or trigonometrically graded materials. The results obtained show that the analysis used to transform the variable coefficients equation into the constant coefficients equation is valid, and the mixed LT-BEM is easy to implement and accurate for finding numerical solutions. The numerical solutions of some test problems are justified by showing their accuracy. Some non-test problems of geometrically symmetric systems are also considered to show the effect of the anisotropy and inhomogeneity of the material on the solutions by verifying the symmetry of solutions. In addition, the effect of boundary conditions is also exhibited.

The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

AbstractWe discuss a semidiscrete analogue of the Unified Transform Method (UTM), introduced by A. S. Fokas, to solve initial‐boundary‐value problems for linear evolution partial differential equations of constant coefficients. The semidiscrete method is applied to various spacial discretizations of several first‐ and second‐order linear equations on the half‐line , producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. We additionally show how the UTM treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and propose the notion of “natural” discretizations. We consider the continuum limit of the semidiscrete solutions and provide several numerical examples.

Estimates for Green's Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients

We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.

On Adomian decomposition method for solving nonlinear ordinary differential equations of variable coefficients

This paper considers the extension of the Adomian decomposition method (ADM) for solving nonlinear ordinary differential equations of constant coefficients to those equations with variable coefficients. The total derivatives of the nonlinear functions involved in the problem considered were derived in order to obtain the Adomian polynomials for the problems. Numerical experiments show that Adomian decomposition method can be extended as alternative way for finding numerical solutions to ordinary differential equations of variable coefficients. Furthermore, the method is easy with no assumption and it produces accurate results when compared with other methods in literature.

Refined mass-critical Strichartz estimates for Schrödinger operators

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

Vortex soliton solutions of a (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential

A (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential is followed with interest. The mapping procedure from the distributed-coefficient to the constant-coefficient equations is provided. Via the procedure with solution of constant-coefficient NLSE from the variational principle, an approximate vortex soliton solution is reported. With the increase of value for topological charge m, torus-shaped vortex soliton turns into ring vortex soliton, and central region of ring gradually enlarges. The layer of vortex solitons and their phase along the z-direction is related to the number of $$q+1$$ with the Hermite parameter q. The branch number of spiral phase structures for vortex solitons is related to the value of topological charge m. The stability test from the direct numerical simulation indicates that vortex solitons with $$m=1,2,3,\, q=0$$ are stable, otherwise, vortex solitons with other values of m, q are unstable.

Lubrication problems solved by the boundary element method

A boundary element method (BEM) for the solution of lubrication problems on finite bearings is presented. The formulation requires the Reynolds equation to be transformed into a constant coefficient equation. Several film shapes that make the transformation possible are systematically obtained. Noticeably, they cover most practical cases. As an example of application, a numerical solution that only requires the discretization of the boundary is presented for a finite pad bearing.