A Comprehensive Study on Symmetry and New Exact Solutions for (3 + 1) – Dimensional Mikhailov-Novikov-Wang Equation

Combined approximations (CA) is an efe cient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations. In previous studies high-quality approximations have been achieved for large changes in the design, but the reasons for the high accuracy were not fully understood. In this work some typical cases, where exact and accurate solutions are achieved by the method, are presented and discussed. Exact solutions are obtained when a basis vector is a linear combination of the previous vectors. Such solutions are obtained also for low-rank modie cations to structures or scaling of the initial stiffness matrix. In general the CA method provides approximate solutions, but the results presented explain the high accuracy achieved with only a small number of basis vectors. Accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent. Such solutions are achieved also for changes in a small number of elements or when the angle between the two vectors representing the initial design and modie ed design is small. Numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges. I. Introduction M ULTIPLErepeatedanalysesareneeded invariousdesign and optimization problems. In general, the structural response cannot be expressed explicitly in terms of the structure properties, and structural analysis involves the solution of a set ofsimultaneous equations. Reanalysis methods are intended to efe ciently analyze structures modie ed due to changes in the design. Approximate reanalysis methods have been used extensively in structural optimization to reduce the number of exact analyses and the overall computational cost during the solution process. The combined approximations (CA) method developed recently is considered in this paper. The method combines several concepts and methods such as reduced basis, series approximations, matrix factorization and Gram ‐Schmidt orthonormalization. These and other methods are used to achieve effective solution procedures. The effectivenessofthemethodinvariousoptimizationproblemshasbeen demonstrated in previous studies. 1i5 Initially the CA method was used only for linear reanalysis models. Recently, the method has been used successfully also in eigenvalue 6 and nonlinear analysis 7 problems. Applications of the method in a large variety ofproblems are discussed elsewhere. 8i11 High-quality approximations of the structural response for large changes in the design have been achieved in previous studies, but the reasons for the high accuracy were not fully understood. In this paper some typical cases, where exact and accurate solutions are achieved by the CAmethod, are presented anddiscussed.In general the CA method provides exact solutions, but the results presented in the paper explain the high accuracy achieved with only a small number of basis vectors.The solution procedure isbriee y described in Sec. II. Three typical cases, where exact solutions are achieved by the CA method, are introduced and discussed in Sec. III. Exact solutions are obtained when a basis vector is a linear combination of the previous vectors. Such solutions are obtained also for lowrank modie cations to structures or scaling of the initial stiffness matrix. Various cases of accurate solutions are discussed in Sec. IV. Convergence properties of the series of basis vectors and the series of the CA terms are presented, and criteria intended to evaluate

Comparison of Exact and Sommerfeld Solution for the Pressure of a Journal Bearing

The Helmholtz-Duffing oscillator is an important vibration model that can be used to study many engineering systems and physical phenomena. In this article, we derived new exact closed-form solutions for the completely integrable Helmholtz Duffing oscillator subject to a constant force. The exact solution was derived naturally from the first integral of the governing differential equation. This approach is completely different from the ansatz method, used in previously published studies, where the exact solution was forced to take the form an initially assumed solution. Through various algebraic transformations and with the aid of standard elliptic integral tables, the present exact period was derived in terms of the complete elliptic integral of the first kind while the exact displacement was derived in terms of the Jacobi cosine function. Unlike the exact ansatz solutions, which have limited range of validity, the present exact solutions are applicable to all bounded periodic states of the Helmholtz-Duffing oscillator with constant force. The validity of the present exact solutions was verified using numerical solutions and published exact solutions in the form of the Weierstrass elliptic function. It was found that the numerical differentiation method produced significant errors for some system inputs and cannot be relied on as a benchmark solution. However, the present exact solutions are benchmark solutions that could be used to check the accuracy of new and existing approximate solutions. Finally, the application of the exact solutions to analyze some real-world problems was demonstrated.

Many phenomena in physics and other fields are often described by nonlinear partial differential equations (NLPDEs). The investigation of exact and numerical solutions, in particular, traveling wave solutions, for NLPDEs plays an important role in the study of nonlinear physical phenomena. These exact solutions when they exist can help one to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations (NLEEs). The ion-acoustic solitary wave is one of the fundamental nonlinear wave phenomena appearing in fluid dynamics [1] and plasma physics [2, 3]. It has recently became more interesting to obtain exact analytical solutions to NLPDEs by using appropriate techniques and symbolical computer programs such as Maple or Mathematica. The capability and power of these software have increased dramatically over the past decade. Hence, direct search for exact solutions is now much more viable. Several important direct methods have been developed for obtaining traveling wave solutions to NLEEs such as the inverse scattering method [3], the tanh-function method [4], the extended tanh-function method [5] and the homogeneous balance method [6]. We assume that the exact solution is expressed by a simple expansion u(x, t) = U(ξ) = ∑ i=0 AiF i(ξ) where Ai are constants to be determined and the function F(ξ) is defined by the solution of an auxiliary ordinary differential equation (ODE). The tanh-function method is the well known method as a direct selection of the function F(ξ) = tanh( ξ). Recently, many exact solutions expressed by various Jacobi elliptic functions (JEFs) of many NLEEs have been obtained by Jacobi elliptic function expansion method [7-10], mapping method [11, 12], F-expansion method [13], extended F-expansion method [14], the generalized Jacobi elliptic function method [15] and other methods [16-20]. Various exact solutions were obtained by using these methods, including the solitarywave solutions, shockwave solutions and periodic wave solutions. The main steps of the F-expansion method [13] are outlined as follows: Step 1. Use the transformation u(x, t) = u(ξ); ξ = k(x−ωt) + ξ0, ξ0 is an arbitrary constant, and reduce a given NLPDE, say in two independent variables,

Exact analytical solutions to simplified cases of nonlinear debris avalanche model equations are necessary to calibrate numerical simulations of flow depth and velocity profiles on inclined surfaces. These problem-specific solutions provide important insight into the full behavior of the system. In this paper, we present some new analytical solutions for debris and avalanche flows and then compare these solutions with experimental data to measure their performance and determine their relevance. First, by combining the mass and momentum balance equations with a Bagnold rheology, a new and special kinematic wave equation is constructed in which the flux and the wave celerity are complex nonlinear functions of the pressure gradient and the flow depth itself. The new model can explain the mechanisms of wave advection and distortion, and the quasiasymptotic front bore observed in many natural and laboratory debris and granular flows. Exact time-dependent solutions for debris flow fronts and associated velocity profiles are then constructed. We also present a novel semiexact two-dimensional plane velocity field through the flow depth. Second, starting with the force balance between gravity, the pressure gradient, and Bagnold’s grain-inertia or macroviscous forces, we construct a simple and very special nonlinear ordinary differential equation to model the steady state debris front profile. An empirical pressure gradient enhancement factor is introduced to adequately stretch the flow front and properly model nonhydrostatic pressure in granular and debris avalanches. An exact solution in explicit form is constructed, and is expressed in terms of the Lambert–Euler omega function. Third, we consider rapid flows of frictional granular materials down a channel. The steady state mass and the momentum balance equations are combined together with the Coulomb friction law. The Chebyshev radicals are employed and the exact solutions are developed for the velocity profile and the debris depth. Similarly, Bagnold’s fluids are also used to construct alternative exact solutions. Many interesting and important aspects of all these exact solutions, their applications to real-flow situations, and the influence of model parameters are discussed in detail. These analytical solutions, although simple, compare very well with experimental data of debris flows, granular avalanches, and the wave tips of dam break flows. A new scaling law for Bagnold’s fluids is established to relate the settlement time of debris deposition. It is found analytically that the macroviscous fluid settles (comes to a standstill) considerably faster than the grain-inertia fluid, as manifested by dispersive pressure.

We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We show that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two corresponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.

Blood flow with body acceleration forces

<div class="section abstract"><div class="htmlview paragraph">Cremer impedance, first proposed by Cremer (Acustica 3, 1953) and then improved by Tester (JSV 28, 1973), refers to the locally reacting boundary condition that can maximize the attenuation of a certain acoustic mode in a uniform waveguide. One limitation in Tester’s work is that it simplified the analysis on the effect of flow by only considering high frequencies or the ‘well cut-on’ modes. This approximation is reasonable for large duct applications, e.g., aero-engines, but not for many other cases of interest, with the vehicle intake and exhaust system included. A recent modification done by Kabral et al. (Acta Acustica united with Acustica 102, 2016) has removed this limitation and investigated the ‘exact’ solution of Cremer impedance for circular waveguides, which reveals an appreciable difference between the exact and classic solution in the low frequency range. Consequently, the exact solution can lead to a much higher low-frequency attenuation level. In addition, the exact solution is found to exhibit some special properties at very low frequencies, e.g., a negative resistance. In this paper, liners designed on the basis of the exact solution are tested and the difference between the exact and classic solution in the low frequency range (not low enough to go into the negative resistance region) is experimentally investigated.</div></div>

The Helmholtz oscillator is a nonlinear mixed‐parity oscillator that models the asymmetric vibrations of many engineering and scientific systems. This paper investigated a general and completely integrable form of the Helmholtz oscillator and derived its exact periodic solution from the first integral of the governing differential equation. The considered Helmholtz oscillator has general linear and nonlinear stiffness constants, is subject to a constant force, and has arbitrary initial conditions. Its exact time period was derived in terms of the complete elliptic integral of the first kind, while the exact displacement was derived in terms of the Jacobi elliptic sine function. The validity of the exact solutions was verified for various combinations of the system parameters and initial conditions by comparing with numerical solutions. The exact solutions and numerical solutions were found to match perfectly. Furthermore, the exact solution was applied to analyze the vibration response of real‐world systems that could be modeled as Helmholtz oscillators. The present exact solutions provide benchmark solutions that could be used to determine the accuracy of new and existing approximate solutions for the Helmholtz oscillator.

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all.

Exact analytical solutions of the Schrödinger equation for a two-dimensional purely sextic double-well potential are proved to exist for a denumerably infinite set of the geometry parameter of the well. First, the geometry values which allow exact solutions are determined. Then, explicit wave functions and corresponding energies are calculated for the allowed geometry values. Concrete exact solutions are given for the principal quantum number n up to 10. Moreover, some interesting rules for the obtained exact analytical solutions are also given; particularly, the number of negative energy levels for a given geometry parameter is obtained. For analyzing the obtained exact solutions and for their classification by quantum number, we also use numerical calculations by the Feranchuk-Komarov operator method.

We present the derivation of an exact special case solution (for a classical lattice) for the Su-Schrieffer-Heeger model for the calculation of soliton dynamics in trans-polyacetylene. Our solution is exact, in the sense that the ansatz state yields an exact solution provided that the equations of motion for its parameters are obeyed. However, these equations can be solved only numerically (in principle to any desired accuracy), not analytically. The model is applied to time simulations of neutral solitons as a function of temperature. We find agreement of the results of our time simulations with experimental data on the mobility of neutral solitons in the system. Comparative calculations using the completely adiabatic model indicate that the results of this model are at variance both with experiment and with those of our model. A simple consideration of the potential barriers for soliton displacement leads to an overestimation of the soliton mobility for low temperatures and an underestimation for higher ones. In an appendix we discuss in some detail the relationship of this exact solution with the state ansatz as presented in our previous paper. We find that the ansatz state and the exact solution yield identical results for lattice momenta, displacements and site occupancies, but differ in a time dependent phase factor. Thus spectra computed with the dynamics resulting from the exact solution for the classical lattice on one hand and from the ansatz state on the other would differ from each other.

In our earlier paper we have shown that the solutions of both the three-dimensional scalar wave equation, which is also the three-dimensional acoustic equation, and Maxwell’s equations have forms in the wave zone, which, except for a factor 1/ r , represent one-dimensional wave motions along straight lines through the origin. We also showed that it is possible to reconstruct the exact solutions from the asymptotic forms. Thus we could prescribe the solutions in the wave zone and obtain the exact solutions that would lead to them. In the present paper we show how the exact solutions can be obtained from the asymptotic solutions and conversely, through the use of a refined Radon transform, which we introduced in a previous paper. We have thus obtained a way of obtaining the exact three-dimensional solutions from the essentially one-dimensional solutions of the asymp­totic form entirely in terms of transforms. This is an alternative way to obtaining exact solutions in terms of initial values through the use of Riemann functions. The exact solutions that we obtain through the use of the Radon transform are causal and therefore physical solutions. That is, these solutions for time t > 0 could have been obtained from the initial value problem by prescribing the solution and its time-derivative, in the acoustic case, and the electric and magnetic fields, in the case of Maxwell’s equations, at time t = 0. The role of time in the relation between the exact solutions and in the asymptotic solutions is made very explicit in the present paper.

This article presents exact algebraic solutions to optimization problems of a double-mass dynamic vibration absorber (DVA) attached to a viscous damped primary system. The series-type double-mass DVA was optimized using three optimization criteria (the H∞ optimization, H2 optimization, and stability maximization criteria), and exact algebraic solutions were successfully obtained for all of them. It is extremely difficult to optimize DVAs when there is damping in the primary system. Even in the optimization of the simpler single-mass DVA, exact solutions have been obtained only for the H2 optimization and stability maximization criteria. For H∞ optimization, only numerical solutions and an approximate perturbation solution have been obtained. Regarding double-mass DVAs, an exact algebraic solution could not be obtained in this study in the case where a parallel-type DVA is attached to the damped primary system. For the series-type double-mass DVA, which was the focus of the present study, an exact algebraic solution was obtained for the force excitation system, in which the disturbance force acts directly on the primary mass; however, an algebraic solution was not obtained for the motion excitation system, in which the foundation of the system is subjected to a periodic displacement. Because all actual vibration systems involve damping, the results obtained in this study are expected to be useful in the design of actual DVAs. Furthermore, it is a great surprise that an exact algebraic solution exists even for such complex optimization problems of a linear vibration system.

A novel connection is uncovered between the simple physics of steady current flow in a composite conductor and the theory of integral equations. With a judicious choice of eigenfunction expansions, exploitation of the physical continuity of current flow across a chosen interface in a composite conductor is shown to yield an infinite class of integral equations with exact closed-form solutions. The mathematical derivation of this class is based on the elementary (but also new) notion of formally equating two different eigenfunction expansions of a given arbitrary function. The new class contains as special cases the celebrated Abel integral equation of classical mechanics and the Kramers–Kronig relations of electromagnetic scattering. But it also contains new integral equations (with exact solutions), some with the Cauchy-singularity 1/(x−y) in their kernels, and a new summation equation. These new equations are in themselves intriguing and their exact solutions do not appear to be derivable by the known methods for solving integral equations. An application of the new class of integral equations is given in the context of a particular composite conductor, which consists of a semi-infinite strip imbedded in an otherwise homogeneous whole space conductor (containing a uniform current flow parallel to the strip). The coefficient in the eigenfunction expansion of the potential in the strip satisfies a one-dimensional singular integral equation with a Cauchy-singularity. This singularity is regularized by the application of an integral equation and its exact solution from the new class, resulting in an integral equation with a smooth kernel. This equation together with the eigenfunction expansion provides an elegant representation for the potential in the strip. (The only known exact solutions are for the cases of elliptic-cylinder and ellipsoid geometries in two and three dimensions, respectively.) The new class of integral equations yields the first examples of singular kernels which possess a bilinear expansion in terms of two different complete sets of eigenfunctions, with only the diagonal terms (i.e., those terms in which the summation indices or integration variables are equal) in the expansion being nonzero. Such an expansion for square-integrable kernels (as opposed to singular kernels) is well known in the Hilbert–Schmidt theory of Hermitian operators and in Schmidt’s extension to the non-Hermitian case, and it forms the basis for a method of solving Fredholm integral equations. None of these theories, however, yields the bilinear expansions for the singular kernels of our new class.