Mikusinski’s operators solution of three-order linear difference equation with variable coefficients

This paper proceeds the papers [3] [4], we make use of the idea of the variable number operators and some concepts and conclusions of the shifting operators series with variable coefficients in the operational field of Mikusinski, it is devoted to the solution of the general three-order linear difference equation with variable coefficients, and it is also devoted to the better solution formula for the some special three-order linear difference equations with variable coefficients: in addition, we try to provide the idea and method for realizing solution of the more than three-order linear difference equation with variable coefficients. Project Supported by the Science Foundation of Anhui Province     

MIKUSINSKI＇S OPERATORS SOLUTION OF THREE－ORDER LINEAR DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

ＭＩＫＵＳＩＮＳＫＩ＇ＳＯＰＥＲＡＴＯＲＳＳＯＬＵＴＩＯＮＯＦＴＨＲＥＥ－ＯＲＤＥＲＬＩＮＥＡＲＤＩＦＦＥＲＥＮＣＥＥＱＵＡＴＩＯＮＷＩＴＨＶＡＲＩＡＢＬＥＣＯＥＦＦＩＣＩＥＮＴＳＺｈｏｕＺｈｉ－ｈｕ（周之虎）（ＡｎｈｕｉＡｒｃｈｉｔｅｃｔｕｒａｌＩ...     

Homogenization of scalar wave equations with hysteresis

The paper deals with a scalar wave equation of the form where is a Prandtl–Ishlinskii operator and are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density and the Prandtl–Ishlinskii distribution function are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data and , where the spatial period tends to 0. We identify the homogenized limits and and prove the convergence of solutions to the solution of the homogenized equation. Received June 17, 1999     

One approximate solution of the Nekrasov problem

An approximate solution ω = A[ω, μ] of the nonlinear integral Nekrasov equation is obtained by successive replacement of the kernel of the integral operator by a close one. The solution is sought not directly at the bifurcation point μ₁ = 3 of the linearized equation ω = μL[ω] but at the point μ = 1 at which operator A[ω, μ], remaining nonlinear in ω, is linear in μ. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 50–56, November–December, 2007.     

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior thatmay vary with time or space. The continuum of order in the fractionalcalculus allows the order of the fractional operator to be considered asa variable. This paper develops the concept of variable and distributedorder fractional operators. Definitions based on the Riemann–Liouvilledefinition are introduced and the behavior of the new operators isstudied. Several time domain definitions that assign different argumentsto the order q in the Riemann–Liouville definition are introduced. Foreach of these definitions various characteristics are determined. Theseinclude: time invariance of the operator, operator initialization,physical realization, linearity, operational transforms, and memorycharacteristics of the defining kernels.A measure (m ₂) for memory retentiveness of the order history isintroduced. A generalized linear argument for the order q allows theconcept of `tailored' variable order fractional operators whose m ₂ memory may be chosen for a particular application. Memory retentiveness (m ₂) andorder dynamic behavior are investigated and applications are shown.The concept of distributed order operators where the order of thetime based operator depends on an additional independent (spatial)variable is also forwarded. Several definitions and their Laplacetransforms are developed, analysis methods with these operators aredemonstrated, and examples shown. Finally operators of multivariable anddistributed order are defined and their various applications areoutlined.     

Exact finite element method

In this paper,a new method,exact element method for constructing finite element,ispresented.It can be applied to solve nonpositive definite or positive definite partialdifferential equation with arbitrary variable coefficient under arbitrary boundarycondition.Its convergence is proved and its united formula for solving partial differentialequation is given.By the present method,a noncompatible element can be obtained and thecompatibility conditions between elements can be treated very easily.Comparing the exactelement method with the general finite element method with the same degrees of freedom,the high convergence rate of the high order derivatives of solution can be obtained.Threenumerical examples are given at the end of this paper,which indicate all results canconverge to exact solution and have higher numerical precision.     

Exact analytic method for solving variable coefficient differential equation

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.     

New truncated expansion method and soliton-like solution of variable coefficient KdV-MKdV equation with three arbitrary functions

The truncated expansion method for finding explicit and exact soliton-like solution of variable coefficient nonlinear evolution equation was described. The crucial idea of the method was first the assumption that coefficients of the truncated expansion formal solution are functions of time satisfying a set of algebraic equations, and then a set of ordinary different equations of undetermined functions that can be easily integrated were obtained. The simplicity and effectiveness of the method by application to a general variable coefficient KdV-MKdV equation with three arbitrary functions of time is illustrated.     

两个新型的八结点块体单元—三维离散算子差分法

给出了弹性力学三维问题的离散算子差分法 ,讨论离散算子差分法在三维问题中的特点 ,意在为该方法的进一步发展提供依据 ,为应用弱形式进行数值求解的研究提供参考。本文从弹性力学平衡方程更为一般的弱形式出发 ,给出了含边界参数的弱形式方程。由该方程不仅可以得到有限元法 ,还可得到离散算子差分法。给出了两个八结点块体单元 ,虽然单元中位移函数是非协调的 ,不需特殊处理便可保证离散格式收敛 ,并对单元位移有十分好的反映能力。     

Approximated-asymptotic solution of the complex variable equation of toroidal shells under axial symmetric loads

The series-approach and the asymptotic-approach are usually used to solve the complex variable equation of the toroidal shells under axial symmetric loads. As is known, the convergence of the series-solution is good only for small values of . On the other hand, the convergence of the asymptotic solution is good only for large values of μ. In this paper, based on an earlier work^[1], a new approach which may be called the approximated-asymptotic solution has been developed and it is valid for both small and large values of μ. It is shown that the results of the approximated-asymptotic solution for toroidal shell with μ=0.5 coincide very well with those of the series-solution, while the results of the asymptotic solution for this value of μ are not as good, and the results of the approximated-asymptotic solution for μ=15 agree with those of the asymptotic solution. This work has been carried out under the direction of Professor Zhang Wei.     

Post-critical behavior of damped beam columns with variable cross section subjected to distributed follower forces

In this paper the post-critical behavior of beam columns with variable mass and stiffness properties subjected to follower forces arbitrarily distributed along their length in the presence of damping (both internal and external) is investigated using a complete nonlinear dynamic analysis. Although the static nonlinear analysis is more economical in computational cost, it is associated only with the loss of local stability via flutter or divergence. Thus, the nonlinear dynamic analysis is adopted in order to examine the global stability of the system. The governing equations of hyperbolic type are derived in terms of the displacements by considering (a) nonlinear response including the axial deformation, (b) nonlinear response excluding the axial deformation and (c) linear response. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a possible FEM solution which may experience a lack of convergence. The problems treated in this investigation include beam columns with various load distributions, such as constant, linear and parabolic. Some of the conclusions detected in studying the nonlinear dynamic stability of Beck’s column with variable cross section (Katsikadelis and Tsiatas, Nonlinear dynamic stability of damped Beck’s column with variable cross section. Int. J. Non-linear Mech. 42, 164–171, 2007), are also valid for the case of distributed loads. The important, however, finding is that the post-critical response under distributed loads depends on the law of distribution of mass and stiffness properties, which may lead also to explosive flutter (unbounded amplitude), in contrast to Beck’s column (end-tip load) where the motion is always bounded.     

Optimization of the ADER‐DG method in GPU applied to linear hyperbolic PDEs

We optimized the Arbitrary accuracy DErivatives Riemann problem (ADER) ‐ Discontinuous Galerkin (DG) numerical method using the CUDA‐C language to run the code in a graphic processing unit (GPU). We focus on solving linear hyperbolic partial–differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behavior is in agreement with the CPU version, but the threshold is slightly larger than in the CPU case. We also observe a big difference when considering single and double precisions where in the first case, the threshold error is significantly larger. Finally, we did observe a speed‐up factor in computational time that depends on the order of the method and the size of the problem. In the best case, our novel GPU implementation runs 23 times faster than the CPU version. We used three partial–differential equation to test the code considering the linear advection equation, the seismic wave equation, and the linear shallow water equation, all of them considering variable coefficients. Copyright © 2015 John Wiley & Sons, Ltd.     

Equivalence Classes and Symmetries of the Variable Coefficient Kadomtsev—Petviashvili Equation

We classify the variable coefficient Kadomtsev—Petviashvili (VCKP) equation into equivalence classes under the group of local point transformations, leaving the equation form invariant but changing the coefficient functions. We list the representatives of all equivalence classes with the corresponding transformations. Then, we obtain the symmetry group of the VCKP equation and in particular discuss how to use these transformations to classify low-dimensional symmetry algebras in the generic case. We conclude with a discussion of the implications of the present article.     

Similarity reductions for 2+1-dimensional variable coefficient generalized kadomtsev-petviashvili equation

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｓｔｕｄｙｏｆｓｈａｌｌｏｗｗａｔｅｒｗａｖｅｓ,ｐｅｏｐｌｅｏｂｔａｉｎｅｄｍａｎｙｗｅｌｌ＿ｋｎｏｗｎｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ,ｍＫｄＶｅｑｕａｔｉｏｎ,Ｂｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎ,Ｗｈｉｔｈａｍ＿Ｂｒｏｅｒ＿ＫａｕｐｅｑｕａｔｉｏｎａｎｄＫｈｏｋｈｌｏｖ＿Ｚａｂｏｌｏｔｓｋａｙａｅｑｕａｔｉｏｎｅｔｃ.[1～8].Ｆｏｒ2 1＿ｄｉｍｅｎｓｉｏｎａｌｖａｒｉａｂｌｅｃｏｅｆｆｉｃｉｅｎｔｇ…     

Isospectral Euler-Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge-Kutta method.     

Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

The Linearized Crocco Equation

In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂_x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way.     

Persoz’s gephyroidal model described by a maximal monotone differential inclusion

Persoz’s gephyroidal model, which consists of elementary rheological models (dry friction element and linear spring), can be covered by the existence and uniqueness theory for maximal monotone operators. Moreover, classical results of numerical analysis allow one to use a numerical implicit Euler scheme, with convergence order of the scheme equal to one. Some numerical simulations are presented.