Auxiliary Linear Equations Research Articles

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Explicit solutions to vertical and horizontal displacements are derived for large deformation of a cantilever beam under point load at the free end by an improved homotopy analysis method (IHAM). Quadratic and cubic nonlinear differential equations are adopted to construct more proficient nonlinear equations for vertical and horizontal displacements respectively combined with their currently available nonlinear displacement equations. Higher-order nonlinear iterative homotopy equations are established to solve the vertical and horizontal displacements by combining simultaneous equations of the constructed nonlinear equations and the auxiliary linear equations. The convergence range of vertical displacement is extended by the homotopy-Páde approximation. The explicit solutions to the vertical and horizontal displacements are in favorable agreements with the respective exact solutions. The convergence ranges for a relative error of 1% by the improved homotopy analysis method for vertical and horizontal displacements increases by 60% and 7%, respectively. These explicit formulas are helpful in practical engineering design for very slender structures, such as high-rise buildings and long bridges.

Positive solutions of BVPs on the half-line involving functional BCs

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations.

Mathematical technology for solving the Kardar-Parisi-Zhang equation with a source

The description of an algorithm for numerical solution of the two-dimensional Kardar-Parisi-Zhang equation with fractal initial condition and spatially inhomogeneous stochastic source of particles has been presented. Using the Cole-Hopf substitution, the KardarParisi-Zhang equation is reduced to an auxiliary linear parabolic equation with a variable coefficient. This reduction provides both compatibility of the algorithm with the concept of mathematical technology and possibility of its practical realization on supercomputer using the MPI parallel programming technology and the ROOT framework. The required profile of the evolving surface is restored by the inverse substitution by solving the auxiliary equation, Opportunity of estimation of its autocorrelation function and its spectral power density, as well as the time dynamics of Shannon surface information, have been discussed.

Matrix Modified Kadomtsev-Petviashvili Hierarchy

Using the bilinear formalism, we consider multicomponent and matrix Kadomtsev-Petviashvili hierarchies. The main tool is the bilinear identity for the tau function realized as the vacuum expectation value of a Clifford group element composed of multicomponent fermionic operators. We also construct the Baker-Akhiezer functions and obtain auxiliary linear equations that they satisfy.

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1−1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 −λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1 $$\tilde U$$ = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution $$\tilde U$$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.

Simulating interaction of nonlinear spatial waves on a free surface of the shallow viscous liquid layer

This paper deals with the combined approach to describing the evolution of weakly nonlinear three-dimensional moderately long perturbations of free surface of viscous liquid. The initial system of hydrodynamic equations is reduced to the novel model system of equations. The first of them is integro-differential equation for nonlinear perturbation of the free surface, taking into account non-stationary shear stress on a weakly sloping bottom. Another equation is an auxiliary linear equation for determining the liquid horizontal velocity vector, averaged over the layer depth. This vector is present in the main equation only in the term of the second order of smallness. The proposed model is suitable for finite-amplitude waves, traveling in different directions in the horizontal plane. Some problems of interactions and collisions of such perturbations over the horizontal and weakly sloping bottom are solved numerically.

Improved Abramov-Petkovšek's Reduction and Creative Telescoping for Hypergeometric Terms

The Abramov-Petkovysek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We improve the Abramov- Petkovysek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The improved reduction does not solve any auxiliary linear difference equation explicitly. It is also more efficient than the original reduction according to computational experiments. Based on this reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. The new algorithm can avoid the costly computation of certificates.

Quasilinear poroelasticity: Analysis and hybrid finite element approximation

We consider a system of partial differential equations which models flows through elastic porous media. This system consists of an elasticity equation describing the displacement of an elastic porous matrix and a quasilinear elliptic equation describing the pressure of the saturating fluid (flowing through its pores). In this model, the permeability depends nonlinearly on the dilatation (divergence of the displacement) of the medium. We show that the solution has regularity. We describe the numerical approximation of solutions using a hybrid finite element‐least squares mixed finite element method. Error estimates are obtained through the introduction of an auxiliary linear elasticity equation. Numerical experiments verify the error estimates and validate the proposed poroelasticity model. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1174–1189, 2015

Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator

A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher- Kolmogorov-Petrovskii-Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions. Symmetry operators for the one-dimensional equation with a constant influence function have been constructed in explicit form and used to obtain a countable set of exact solutions. The symmetry properties of a differential equation (DE) are associated with transforms leaving invariant the set of solutions of the equation. We shall refer to these transforms as symmetry operators (SOs). Symmetry operators make it possible to generate new solutions of an equation by using them sequentially to act on a known solution. The solution generation procedure only illustrates the potentialities of symmetry operators, but does not exhaust them. The problem is how to find symmetry operators or other symmetry transforms in explicit form. The basic mathematical apparatus used in a comprehensive study of symmetries of differential equations is the Lie group theory of continuous transformations leaving an (ordinary or partial) differential equation invariant. The invariance group of an equation is also termed a symmetry group, or a group admitted by the equation. Seeking a symmetry group is reduced to solving a system of linear constitutive equations for an infinitesimal Lie group operator (generator). The Lie group of finite transformations constructed with a generator can be applied to any solution of the equation admitting the group, and thus generate parametric sets of new solutions to the equation from a known one. A Lie group of pointwise invariance transformations of a differential equation can be considered as a set of symmetry operators of the equation. A detailed description of the application of Lie groups to ordinary DEs can be found, for instance, in Refs. 1-3. For partial differential equations (PDEs), the application of Lie group methods leans upon a procedure of prolongation of the action of a Lie group on higher partial derivatives (1-3). The generator of a prolonged Lie group has special structure and is a base object of examination in a study of the PDE symmetry properties. For a Lie group admitted by a PDE, the generator is determined by a linear partial differential equation in the generator coefficients that specify the so-called symmetries of the equation. A set of symmetries possesses algebraic properties, which are used to examine the properties of equations and to find sets of their solutions (1-3). However, the problem of finding SOs of nonlinear equations in explicit form by direct methods not related to Lie groups is poorly studied. In general statement, this problem has no constructive solution due to significant

Two classes of positive solutions of first order functional differential equations of delayed type

The equation ẏ(t)=−f(t,yt) is considered. The existence of two classes of positive solutions asymptotically different for t→+∞ is proved using the retract method. With the aid of solutions of two auxiliary linear equations with several delays, constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.

The multicomponent KP hierarchy: differential Fay identities and Lax equations

In this paper, we show that four sets of differential Fay identities of an N-component KP hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wavefunctions. From this, we derive the Lax representation for the N-component KP hierarchy, which are equations satisfied by some pseudo-differential operators with matrix coefficients. Besides the Lax equations with respect to the time variables proposed in Date et al (1981 J. Phys. Soc. Japan 50 3806–12), we also obtain a set of equations relating different charge sectors, which can be considered as a generalization of the modified KP hierarchy proposed in Takebe (2002 Lett. Math. Phys. 59 157–72).

A New Optimal Control Formulation to Ensure the Stability of NMPC Systems

AbstractNonlinear model predictive control (NMPC) has been considered as a promising control algorithm, since a rigorous model can be explicitly employed and state variable restrictions ensured by formulating inequality constraints. However, due to the nonlinear problem formulation, it is difficult to analyze the stability properties of NMPC systems. In this study we propose a new formulation of the optimal control problem to ensure the stability of NMPC systems. Auxiliary state variables and linear state equations will be introduced and heir eigenvalues optimized in the dynamic optimization framework. System state variables will be constrained by these auxiliary variables so that they will conform to the stability properties of the auxiliary variables. Features of this stabilization approach are analyzed. Results of case studies indicate satisfactory effectiveness of the proposed NMPC.

Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type

Solutions of the equation −f(t, yt) are considered for t → ∞. The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin′s technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right‐hand side of the equation considered, inequalities for both types of positive solutions are given as well.

Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.

Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multicomponent KP hierarchy. To this end, the multicomponent KP hierarchy is (re)formulated to depend on several discrete variables called “charges”. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multicomponent version of the so called differential Fay identity are derived from the Hirota equations of the τ -function of this “charged” multicomponent KP hierarchy. These multicomponent differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton–Jacobi equations for the S -functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multicomponent KP hierarchy.

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g , the standard universal solution R ( x ) ∈ U q ( g ) ⊗ 2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R-matrix and from the solution F ( x ) ∈ U q ( g ) ⊗ 2 of the Quantum Dynamical coCycle Equation as R ( x ) = F 21 −1 ( x ) R F 12 ( x ) . F ( x ) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation. Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g = sl ( n + 1 ) ( n ⩾ 1 ) only, there could exist an element M ( x ) ∈ U q ( sl ( n + 1 ) ) such that the dynamical gauge transform R J of R ( x ) by M ( x ) , R J = M 1 ( x ) −1 M 2 ( x q h 1 ) −1 R ( x ) M 1 ( x q h 2 ) M 2 ( x ) , does not depend on x and is a universal solution of the Quantum Yang–Baxter Equation. In the fundamental representation, R J corresponds to the standard solution R for n = 1 and to Cremmer–Gervais's one R 12 J = J 21 −1 R 12 J 12 for n > 1 . For consistency, M ( x ) should therefore satisfy the Quantum Dynamical coBoundary Equation, i.e. F ( x ) = Δ ( M ( x ) ) J M 2 ( x ) −1 ( M 1 ( x q h 2 ) ) −1 , in which J ∈ U q ( sl ( n + 1 ) ) ⊗ 2 is the universal cocycle associated to Cremmer–Gervais's solution. The aim of this article is to prove this conjecture and to study the properties of the solutions of the Quantum Dynamical coBoundary Equation. In particular, by introducing new basic algebraic objects which are the building blocks of the Gauss decomposition of M ( x ) , we construct M ( x ) in U q ( sl ( n + 1 ) ) as an explicit infinite product which converges in every finite dimensional representation. We emphasize the relations between these basic objects and some non-standard loop algebras and exhibit relations with the dynamical quantum Weyl group.

Dispersionless Hirota Equations of Two-Component BKP Hierarchy

The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dis- persionless limit of this multi-component hierarchy is considered on the level of the -func- tion. The so called dispersionless Hirota equations are obtained from the Hirota equations of the -function. These dispersionless Hirota equations turn out to be equivalent to a sys- tem of Hamilton-Jacobi equations. Other relevant equations, in particular, dispersionless Lax equations, can be derived from these fundamental equations. For comparison, another approach based on auxiliary linear equations is also presented.

Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations

We consider the calibration of parameters in physical models described by partial differential equations. This task is formulated as a constrained optimization problem with a cost functional of least squares type using information obtained from measurements. An important issue in the numerical solution of this type of problem is the control of the errors introduced, first, by discretization of the equations describing the physical model, and second, by measurement errors or other perturbations. Our strategy is as follows: we suppose that the user defines an interest functional I, which might depend on both the state variable and the parameters and which represents the goal of the computation. First, we propose an a posteriori error estimator which measures the error with respect to this functional. This error estimator is used in an adaptive algorithm to construct economic meshes by local mesh refinement. The proposed estimator requires the solution of an auxiliary linear equation. Second, we address the question of sensitivity. Applying similar techniques as before, we derive quantities which describe the influence of small changes in the measurements on the value of the interest functional. These numbers, which we call relative condition numbers, give additional information on the problem under consideration. They can be computed by means of the solution of the auxiliary problem determined before. Finally, we demonstrate our approach at hand of a parameter calibration problem for a model flow problem.

Modeling epithelial cell homeostasis: assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.