Nonlinear Forcing Term Research Articles

Burgers equation with time-dependent coefficients and nonlinear forcing term: Linearization and exact solvability

We construct and discuss a linearization method for solving Burgers equation with time-dependent coefficients and a nonlinear forcing term. Our results are shown to contain and generalize recent findings (Miskinis, 2001; Buyukasik and Pashaev, 2013). As applications of our method we solve several initial- and boundary-value problems for Burgers equation with forcing of sinusoidal, polynomial, as well as X1-Laguerre exceptional orthogonal polynomial type.

Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation

A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N + 1)(N + 2)/2 − 3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N ≤ K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.

Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.

Third harmonic shear horizontal and Rayleigh Lamb waves in weakly nonlinear plates

The third order harmonic generation (third harmonics as well as cubic sum and difference harmonics) due to the cubic interaction of two collimated elastic waves in a homogeneous, isotropic, weakly nonlinear plate is investigated by using a fourth order expansion of strain energy density to formulate the nonlinear boundary problems. Waves with both shear horizontal (SH) and Rayleigh Lamb (RL) nature are considered as primary or tertiary wave fields. The non-zero power flux condition is evaluated using characteristic parity matrices of the cubic nonlinear forcing terms and third order harmonic mode shapes. Results indicate that waves with either SH or RL nature receive power flux from a specific pattern of primary mode interaction. Further analytical evaluation of the synchronism condition enables identification of primary SH and RL modes that are able to generate cumulative third harmonics. The primary SH modes are shown to be holo-internal-resonant with third harmonic SH fields. This simply means that all points on the primary dispersion curves are internally resonant with third harmonics, which is not the case for second harmonics. Such flexibility will be advantageous for laboratory and field measurements.

Interaction of torsional and longitudinal guided waves in weakly nonlinear circular cylinders

The nonlinear forcing terms for the wave equation in general curvilinear coordinates are derived based on an isotropic homogeneous weakly nonlinear elastic material. The expressions for the nonlinear part of the first Piola-Kirchhoff stress are specialized for axisymmetric torsional and longitudinal fundamental waves in a circular cylinder. The matrix characteristics of the nonlinear forcing terms and secondary mode wave structures are manipulated to analyze the higher harmonic generation due to the guided wave mode self-interactions and mutual interactions. It is proved that both torsional and longitudinal secondary wave fields can be cumulative by a specific type of guided wave mode interactions. A method for the selection of preferred fundamental excitations that generate strong cumulative higher harmonics is formulated, and described in detail for second harmonic generation. Nonlinear finite element simulations demonstrate second harmonic generation by T(0,3) and L(0,4) modes at the internal resonance points. A linear increase of the normalized modal amplitude ratio A2/A1(2) over the propagation distance is observed for both cases, which indicates that mode L(0,5) is effectively generated as a cumulative second harmonic. Counter numerical examples demonstrate that synchronism and sufficient power flux from the fundamental mode to the secondary mode must occur for the secondary wave field to be strongly cumulative.

Modeling of Nonlinear, Spatially-Dispersive Plasmas and Semiconductors Under Harmonic Excitation

The nonlinear, spatially-dispersive response of a semiconductor or plasma to large-amplitude time-harmonic electromagnetic fields is obtained by solving the nonlinear transport equation using an harmonic expansion. The conduction response, which is nonlinear and generally spatially and temporally dispersive, is given as a hierarchical set of linear second-order differential equations with non-linear forcing terms. The polarization response is assumed linear. A simple slab example is shown that admits analytical solutions for the nonlinear material response to various orders. As the solution order grows, the nonlinear forcing terms grow in complexity, although the differential equations remain second-order. In the static limit, the two lowest-order solutions are shown to identically satisfy the dc transport equation.

Admissibly inertial manifolds for a class of semi-linear evolution equations

Consider the semi-linear evolution equation du(t)dt+Au(t)=f(t,u). We prove the existence of a new class of inertial manifolds called admissibly inertial manifolds for this equation. These manifolds are constituted by trajectories of the solutions belonging to rescaledly admissible function spaces which contain wide classes of function spaces like weighted Lp-spaces, the Lorentz spaces Lp,q and many other rescaling function spaces occurring in interpolation theory. The existence of these manifolds is obtained in the case that the partial differential operator A is positive definite and self-adjoint with a discrete spectrum, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions on the domain D(Aθ), 0⩽θ<1, i.e., ‖f(t,x)−f(t,y)‖⩽φ(t)‖Aθ(x−y)‖ and ‖f(t,x)‖⩽φ(t)(1+‖Aθx‖) for φ(⋅) being a real and positive function which belongs to certain classes of admissible function spaces.

Existence and Stability of Almost Periodic Solutions of Nonlinear Damped Equations of a Suspended String

In this paper we shall show the existence and the stability of almost periodic solutions of the boundary value problem to a nonlinear suspended string equation with a linear damping term and an almost periodic weakly nonlinear forcing term. We treat both weak solutions and strong solutions. Also we show the existence of time global solutions of the initial boundary value problem to the equation.

On the domain of analyticity of solutions to semilinear Klein–Gordon equations

We consider initial-value problems for semilinear Klein–Gordon equations u t t − Δ x u + u + f ( u ) = 0 with periodic boundary conditions. Assuming that both the initial data and the nonlinear forcing term f ( u ) are analytic, we provide explicit lower bounds on the decay of the radius ρ ( t ) of analyticity of the solutions as a function of time. In particular, in one space dimension, with u real valued and f ( u ) = u 2 k + 1 , we prove that the decay of ρ ( t ) is not worse than 1 / t . The results are given in a general framework, including Gevrey class solutions.

Fourth-order compact difference and alternating direction implicit schemes for telegraph equations

In this paper, two- and three-level compact difference and alternating direction implicit schemes are presented for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The stability and error estimates are given. The convergence rates of the present schemes are of order O ( τ 2 + h 4 ) . Numerical experiments on model problems show that the present schemes are of high accuracy.

On nonlocal three-point boundary value problems of Duffing equation with mixed nonlinear forcing terms

In this paper, we investigate the existence and approximation of the solutions of a nonlinear nonlocal three-point boundary value problem involving the forced Duffing equation with mixed nonlinearities. Our main tool of the study is the generalized quasilinearization method due to Lakshmikantham. Some illustrative examples are also presented. Mathematics Subject Classification (2000): 34B10, 34B15.

Electrostatic end-field defocusing of neutral atoms and its compensation

Neutral atoms entering an electric field experience a defocusing force in the dipole field direction, which is proportional to the field gradient. If an experiment, such as the search for a permanent electron electric dipole moment (eEDM), requires a very strong electric field ($13.5\text{ }\text{ }\mathrm{MV}/\mathrm{m}$), then this end-field defocusing results in beam blowup and much reduced phase-space acceptance. In this paper we discuss how these defocusing fields arise from the longitudinal changes in the electric dipole field and their dependence on the electrode shape and spacing between lenses. We find that the end-field defocusing comes from strong impulse forces, whose defocusing power was calculated for simple electrodes with rounded ends. To compensate for this end-field defocusing, a triplet of transverse-focusing lenses was added to the pure dipole field plates in the generic eEDM cesium fountain experiment used to study the neutral beam optics. Envelope equations, which calculated the beam sizes of the atom bunch for the linear forces, are used to obtain a set of lens parameters that give a well focused beam in the fountain. Atom trajectory equations allow us to calculate the phase-space acceptance of the lens system with the nonlinear force terms included.

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson–Boltzmann equation and diffusion–reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.

Adaptive nonlinear dynamical initialization

[1] An adaptive nonlinear initialization scheme is presented that provides improvements over linear relaxation methods for approaching the target state while minimizing discontinuities in dynamical models. The method adds an adaptive nonlinear forcing term to the model equations that is a function of the difference between the model field and its target value. A new feature is that the amplitude of the forcing is nonlinearly adapted to the size of the difference, allowing for stronger relaxation where differences are large and weaker relaxation where differences are small. We find that the function leads to more optimal introduction of new information by working hardest at the beginning of the initialization period while converging toward a steady condition for the majority of the domain at the end of the initialization period. Experiments with a limited area ocean model, with different dynamical regimes, show that the adaptive scheme leads to less shock than standard linear approaches and permits the model to converge to a state away from the target field if the target is not a priori dynamically balanced. Results indicate that the method has the potential to lower forecast error. This suggests that it will have a broad range of applications in dynamical prediction systems.

Fully nonlinear boundary value problems with impulse

An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth as t → ∞ or t → 0+. The nonlinear impulse terms and the nonlinear boundary terms are assumed to satisfy the analogous asymptotic behavior.

Boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term

In this paper, we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing term x ″ + f ( x ) g ( x ′ ) + γ | x | α − 1 x = p ( t ) , where f ( x ) , p ( t ) are odd functions, p ( t ) is smooth 1-periodic function and γ ≠ 0 is a constant. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, we show the existence of Aubry–Mather sets as well.

NUMERICAL SOLUTIONS FOR SPACE FRACTIONAL DISPERSION EQUATIONS WITH NONLINEAR SOURCE TERMS

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order <TEX>$O({\Delta}x+{\Delta}t)$</TEX> is obtained in the discrete <TEX>$L_2$</TEX> norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

A study of the polarization–electric-field nonlinear effect on the response of smartcomposite plates

Polarization–electric-field (P–E) interaction results in rendering the stress tensor non-symmetric and in a nonlinear forceterm in the equilibrium equation. In this paper, an attempt is made to study these (P–E) nonlinear effects on the static response of laminated composite plates with piezo actuators.Further, this paper also focuses on finding the most effective piezo lay-up and plyorientation which gives the maximum deflections. Four different piezo lay-up configurationsand three ply orientations are considered. It has been observed from the study thatwidth-wise strips show more transverse bending and twisting. However, full length piezolayers show maximum longitudinal bending. The results of nonlinear analysis show a moreconsiderable softening trend in deformations than that of the linear analysis in the case oflongitudinal bending and twisting. In the case of transverse bending this nonlineareffect shows a hardening trend. Further, it has been observed that the influence ofP–E nonlinearity depends on the stiffness of the core material, the geometric arrangement ofpiezo patches, the boundary conditions and the actuation voltage.

SECOND-ORDER METHOD FOR SOLVING 2D NONLINEAR PARABOLIC DIFFERENTIAL EQUATIONS BASED ON ADI METHOD

In this article, a new unconditionally stable method based on alternating direction implicit (ADI) method for solving nonlinear unsteady 2D parabolic problems is presented. The order of this method in both time and space is 2. For this development, the Peaceman–Rachford ADI method has been modified suitably to take care of nonlinear forcing term of the equation appropriately. The unconditional stability of the method is shown under discrete L2 norm on bounded sets. The method of solution consists of a number of strictly diagonally dominant tridiagonal matrices, which make the method computationally efficient. The accuracy of the proposed method is also demonstrated by some numerical examples.

Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U = ( U ( t , s ) ) t ⩾ s ⩾ 0 on a half-line R + and a semi-linear integral equation u ( t ) = U ( t , s ) u ( s ) + ∫ s t U ( t , ξ ) f ( ξ , u ( ξ ) ) d ξ . We prove the existence of stable manifolds of solutions to this equation in the case that ( U ( t , s ) ) t ⩾ s ⩾ 0 has an exponential dichotomy and the nonlinear forcing term f ( t , x ) satisfies the non-uniform Lipschitz conditions: ‖ f ( t , x 1 ) − f ( t , x 2 ) ‖ ⩽ φ ( t ) ‖ x 1 − x 2 ‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p , q and many other function spaces occurring in interpolation theory.