Quasi-exact Solutions Research Articles

Nonequilibrium response of magnetic nanoparticles to time-varying magnetic fields: Contributions from Brownian and Néel processes.

Many technical and biomedical applications of magnetic nanoparticles rely on their response to time-varying magnetic fields. While well-established models exist for either immobile or thermally blocked nanoparticles, the intermediate regime where Brownian as well as Néel relaxation occur at the same time is less well explored. Here, we use an efficient model that allows us to study the nonlinear dynamics of individual magnetic nanoparticles in response to different time-varying magnetic fields over a broad range of field parameters, taking into account both relaxation mechanisms. We provide quasiexact solutions for the longitudinal dynamics as well as approximate formulas from dynamic mean-field theory. Our results are relevant, e.g., for magnetorelaxometry, magnetic fluid hyperthermia, and magnetic particle imaging. For these example applications, we show that the ratio of characteristic Brownian to Néel relaxation time can have a profound impact on characteristic response quantities, especially at large field strengths.

The torsional rigidity of an orthotropic bar of unidirectional composite laminate part I: Theoretical solution

In 1934, Timoshenko derived and published the torsional rigidity of a rectangular bar of isotropic material by using the membrane analogy. The rigidity depends on the bar’s shear modulus, width and thickness. In 1950, Lekhnitskii followed Timoshenko’s process to derive the torsional rigidity for an orthotropic bar of unidirectional composite laminate, where the rigidity depends on the laminate’s width, the thickness and the shear moduli. In 1990, Tsai, Daniel and Yaniv solved the same case, deriving a quasi-exact solution of torsional rigidity. In the same year, Tsai and Daniel verified their result through multiple experiments. All these rigidities become different when the definitions of thickness and width are swapped. However, they remain identical numerically, lacking mathematic proof for nearly a century. In 2022, Tsai et al. resolved Timoshenko’s case by considering all the conditions and energy minimization criterion. By a completely different approach, the torsional rigidity was derived in a completely different form from that of Timoshenko but numerically identical. Moreover, the solution satisfies the rule of swapping, which makes perfect sense physically. This work applies Tsai’s process to Lekhnitskii’s case. A general solution is derived that satisfies the rule of swapping involving the swapping of shear moduli.

Fully discrete WENO with double entropy condition for hyperbolic conservation laws

ABSTRACTThis paper put forward a new fully discrete scheme construction method – double entropy condition solution formula method. With that, we turn the state-of-the-art semi-discrete WENO + RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work , the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems.

Exact and quasi-exact Newton dynamics of n − 1 and n bodies n − 1 of which have equal masses

For n,n≥3 gravitating bodies n−1 of which have equal masses planar exact and spatial quasi-exact solutions of the Newton equation of motion are found. Exact planar solutions are also found for this equation for n−1,n≥3 bodies with equal masses. These solutions describe a motion of the bodies with equal masses in such a way that their coordinates coincide with vertices of a regular polygon in horizontal planes while the remaining body is either immobile at the origin or moves along the vertical axis crossing the origin. The Weinstein and center Lyapunov theorems are utilized.

Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method

The current article is concerned with a comprehensive investigation of achieving the simplest solution of non-conservative coupled nonlinear forced oscillators. The article mainly depends on a non-perturbative method. It depends on yielding an equivalent linear system. The advantage of this linear system is that its coefficients are easily computed and that they include the effects of the original nonlinear coefficients. This linearization approach allows achieving quasi-exact solutions even in the presence of periodic forces. The relationships of frequencies with amplitudes are easily established. The analytical solution so yielded may serve as a basis for a qualitative understanding of the actual behavior of the coupled nonlinear oscillators. Additionally, numerical calculations are carried out graphically to address the validation of the new approach and to further illustrate the effectiveness and convenience of the method. The results are compared with the exact numerical solutions which show perfect accuracy. The method can be easily extended to other nonlinear systems and can therefore be exceedingly applicable in engineering and other sciences.

Quasi-exact solutions for guided modes in two-dimensional materials with tilted Dirac cones

We show that if the solutions to the (2+1)-dimensional massless Dirac equation for a given one-dimensional (1D) potential are known, then they can be used to obtain the eigenvalues and eigenfunctions for the same potential, orientated at an arbitrary angle, in a 2D Dirac material possessing tilted, anisotropic Dirac cones. This simple set of transformations enables all the exact and quasi-exact solutions associated with 1D quantum wells in graphene to be applied to the confinement problem in tilted Dirac materials such as 8-Pmmn borophene. We also show that smooth electron waveguides in tilted Dirac materials can be used to manipulate the degree of valley polarization of quasiparticles travelling along a particular direction of the channel. We examine the particular case of the hyperbolic secant potential to model realistic top-gated structures for valleytronic applications.

Quasi-exact solution of the anharmonic oscillator in curved space–time with tensor potential of type Ar + B/r3 and spin and pseudo-spin symmetries

In this work, we analyzed the anharmonic oscillator with spin and pseudo-spin symmetries in deformed nuclei. For that, we consider the Dirac equation in curved space–time which has a line element given by [Formula: see text] with electromagnetic field [Formula: see text]. We consider two forms of coupling of the spin 1/2 particle with the electromagnetic field and curved space–time: [Formula: see text] and [Formula: see text], where the spin and pseudo-spin symmetries were manifested, respectively. We calculate the spinorial wave function and the energy spectrum of the anharmonic oscillator quasi-exactly, and from this result it is obtained that the symmetries are broken due to the coupling of the electromagnetic field with the curvature of space–time. We analyzed the densities of radial probabilities and energy spectra in both symmetries.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

Quasi-exact solution of the Dirac equation on curved space-time with Coulomb scalar and vector potentials and Mie-type tensor potential with pseudo-spin and spin symmetries

In this paper we solve the Dirac equation with the Coulomb scalar U(r) and vector V(r) potentials and type-Mie tensor potential in curved space-time whose metric is of type with spherical symmetry. For this we consider two types of symmetry in the system, first spin symmetry with V(r) = U(r) and then with pseudo-spin symmetry V(r) = − U(r). In both cases we have a tensor potential A(r) that coupling through the electromagnetic field A μ = (V(r), cA(r), 0, 0). We find, quasi-exactly, the Dirac spinor and the energy spectrum of the system, and we sketch the probability densities and some energy spectra.

A quasi-exact solution for the analysis of smart multilayered simply supported shallow shell panels

A quasi-three-dimensional solution for the bending analysis of simply supported orthotropic piezoelectric shallow shell panels subjected to thermal, mechanical and electrical potential is introduced. The mechanical governing equations are derived in term of three-dimensional equilibrium relations and the classical Maxwell’s equations. The trough-the-thickness temperature is modeled by the Fourier’s heat conduction equation. The coupled partial differential equations are solved by Navier closed form solutions. The trough-the-thickness profile for electrical potential, temperature profile and displacements is obtained by using a quasi-exact method so-called the differential quadrature method (DQM). Chebyshev polynomials of the third kind are used as the basis functions and the grid thickness domain discretization for the DQM. The interlaminar conditions for transverse stresses, temperature and electrical potential are imposed. The correct traction conditions for transverse stresses and scalar potential function at the top and the bottom are applied. The results for cylindrical, spherical and rectangular plates are presented. The excellent obtained results are compared with layerwise and three-dimensional solutions reported in the literature.

Error estimation for 2-D crack analysis by utilizing an enriched natural element method

This paper presents an error estimation technique for 2-D crack analysis by an enriched natural element (more exactly, enriched Petrov-Galerkin NEM). A bare solution was approximated by PG-NEM using Laplace interpolation functions. Meanwhile, an accurate quasi-exact solution was obtained by a combined use of enriched PG-NEM and the global patch recovery. The Laplace interpolation functions are enriched with the near-tip singular fields, and the approximate solution obtained by enriched PG-NEM was enhanced by the global patch recovery. The quantitative error amount is measured in terms of the energy norm, and the accuracy (i.e., the effective index) of the proposed method was evaluated using the errors which obtained by FEM using a very fine mesh. The error distribution was investigated by calculating the local element-wise errors, from which it has been found that the relative high errors occurs in the vicinity of crack tip. The differences between the enriched and non-enriched PG-NEMs have been investigated from the effective index, the error distribution, and the convergence rate. From the comparison, it has been justified that the enriched PG-NEM provides much more accurate error information than the non-enriched PG-NEM.

Quasi-exact and asymptotic iterative solutions of Dirac equation in the presence of some scalar potentials

In this paper, the Dirac equation in the presence of some scalar potentials based on sl(2) Lie algebra is solved by quasi-exact solvability theory. The configuration of the classes III and VI potentials in the Turbiner’s classification is constructed. Then, the Bethe ansatz equations are calculated so that the energy eigenvalues and eigenfunctions are obtained. Also, we study the problem by using asymptotic iteration method. Finally, we compare the results obtained by these two methods.

An error-driven grid refinement for 2-D reliable crack analysis by enriched natural element method

An error-driven grid refinement technique is introduced for 2-D reliable crack analysis by an enriched natural element method (more exactly, Petrov-Galerkin natural element method). A quasi-exact solution for a posteriori error estimation was obtained by enhancing the bare approximation solution of NEM (natural element method) using the enrichment method and the global patch recovery. The proposed method is illustrated through the error-driven grid refinement for a rectangular plate with a slant edge crack. The quantitative error amount is measured in terms of the energy norm, and the accuracy (i.e., the effective index) of the proposed method was evaluated from the comparison with the errors which were obtained by FEM using a very fine mesh. The proposed method provides the effective index which is much improved from that of non-enriched PG-NEM. The NEM grid was non-uniformly refined based on the local error information, and the resulting error distributions were investigated. It has been observed that the difference between the maximum and minimum values in the local error distribution of enriched PG-NEM is larger than that of non-enriched PG-NEM. The reduction of global errors according to the non-uniform grid refinement was also investigated to the grid density, from which it is found that the enriched PF-NEM provides smaller errors than the non-enriched PG-NEM. In addition, the proposed grid refinement provides the 1.676 times higher convergence rate than the uniform grid refinement.

Quasi-exact solutions for the Bohr Hamiltonian with sextic oscillator potential

A discussion on the quasi-exact solution of the Bohr Hamiltonian with sextic oscillator potential is made by attracting the attention on some recent results of its application to the phase transition from spherical vibrator to a γ-unstable system. More precisely, it is underlined the importance of the solvability order on the structure of the states, especially in the critical point, respectively, in the deformed region of the phase transition.

Quasiexact Theory of Three-Dimensional Optical Self-Focusing

We find a quasiexact three-dimensional analytic solution of the nonlinear Schrodinger equation describing the field of a stationary optical beam in an unbounded homogeneous nonlinear isotropic medium supporting a state of linear polarization.

A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

Quasi-exact solution of the Riemann problem for generalised dam-break over a mobile initially flat bed

This paper investigates dam-break problems with flows on one or two sides of zero or nonzero velocities over a mobile initially flat bed, and quasi-exact solutions are presented by solving the Riemann problems using the simple wave theory. The flow structures after dam collapse for nonzero velocities are much richer than those for zero velocities on both sides, although they are also a combination of waves of different characteristic families, which are consistent with Lax [CBMS-Regional Conference Series in Applied Mathematics, SIAM, 1973]. The wave can be a rarefaction, a shock, or a combination of a rarefaction and a semi-characteristic shock. The semi-characteristic shock is related to the morphodynamic characteristics. The relationship between morphodynamic and hydrodynamic characteristics is illustrated, along with types of waves (shock, rarefaction or a combination of these), and sediment convergence and type of characteristic. It is shown that the types of waves that may occur in the Riemann solution, and, in some cases, their possible approximate locations, can be determined prior to the construction of the Riemann solution itself. The Riemann solution presented here can be used to study shock–shock interactions.

Perturbed Coulomb Potentials in the Klein–Gordon Equation: Quasi-Exact Solution

Using the Lie algebraic approach, we present the quasi-exact solutions of the relativistic Klein–Gordon equation for perturbed Coulomb potentials namely the Cornell potential, the Kratzer potential and the Killingbeck potential. We calculate the general exact expressions for the energies, corresponding wave functions and the allowed values of the parameters of the potential within the representation space of sl(2) Lie algebra. In addition, we show that the considered equations can be transformed into the Heun’s differential equations and then we reproduce the results using the associated special functions. Also, we study the special case of the Coulomb potential and show that in the non-relativistic limit, the solution of the Klein–Gordon equation converges to that of Schrodinger equation.

A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates

The free flexural vibration of thin rectangular plates is revisited. A new, quasi‐exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam‐like expressions. Using the proposed quasi‐exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB®, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS® and in‐house conventional FEM programs using both 12‐ and 16‐DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi‐exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two‐dimensional structural components during the preliminary design stages, before a detailed design begins.

A numerical method based on Taylor series for bifurcation analyses within Föppl–von Karman plate theory

A new numerical technique for post-buckling analysis is presented by combining the asymptotic numerical method (ANM) and the Taylor meshless method (TMM). These two methods are based on Taylor series, with respect to a scalar load parameter for ANM, and with respect to the space variables for TMM. The advantage of ANM is an adaptive step length, which is very efficient near bifurcation points. The specificity of TMM is a quasi-exact solution of the partial differential equations inside the domain, which leads to a strong reduction of the number of degrees of freedom.