Nonlinear Differential Problem Research Articles

A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations

Optimal homotopy asymptotic method (OHAM) is prolifically implemented to find the optimal solutions of fractional order heat- and wave-like equations. We inspect the competence of the method by examining fractional order time dependent partial differential equations. It is observed that OHAM is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective and easy to use, for handling more general fractional order heat- and wave-like models.

Existence of solutions to bilinear problems with a closed-loop control

Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control z + A (z ) = B (u,z ) + f, u ∈ U (t,z ),z (0) = z 0 where the operator B is bilinear with respect to the control u and the state z in reflexive, separable Banach spaces denoted Y and V , respectively. The operator A is nonlinear in V , and given a positive real number T , the set-valued map U is defined in [ 0,T ] × V . Without making any assumptions about the convexity of U , its values are taken to be non-empty closed, decomposable subsets of Y .

Comparison between Spectral perturbation and Spectral relaxation approach for unsteady heat and mass transfer by MHD mixed convection flow over an impulsively stretched vertical surface with chemical reaction effect

In this study, the spectral perturbation method (SPM) is utilized to solve the momentum, heat and mass transfer equations describing the unsteady MHD mixed convection flow over an impulsively stretched vertical surface in the presence of chemical reaction effect. The governing partial differential equations are reduced into a set of coupled non similar equations and then solved numerically using the SPM. The SPM combines the standard perturbation method idea with the Chebyshev pseudo-spectral collocation method. In order to demonstrate the accuracy and efficiency of the proposed method, the spectral perturbation (SPM) numerical results are compared with numerical results generated using the spectral relaxation method (SRM) and a good agreement between the two methods is observed up to a minimum of eight decimal digits. Several simulation are conducted to ascertain the accuracy of the SPM and the SRM. The computational speed of the SPM is demonstrated by comparing the SPM computational time with the SRM computational time. A residual error analysis is also conducted for the SPM and the SRM in order to further assess the accuracy of the SPM. The study shows that the spectral perturbation method (SPM) is more efficient in terms of computational speed when compared with the SRM. The study also shows that the SPM can be used as an efficient and reliable tool for solving strongly nonlinear boundary value partial differential equation problems that are defined under the Williams and Rhyne [3] transformation. In addition, the study shows that accurate results can be obtained using the perturbation method and thus, the conclusions earlier drawn by researchers regarding the accuracy of perturbation methods is corrected.

Results for nonlinear fractional differential equations on pseudo-metric spaces with three boundary val

Abstract. In this paper, we investigate the existence of solutions in Gauge spaces for the nonlinear fractional differential equation boundary value problem With different boundary value and and different order 1 < a 2 and 0 < a < 1. where 0 < /31 < /32 < T and µ [O, T]and + is the standard Riemann-Lowville differentiation with f(t, u(t)) C([O, T] x [O, ), R). By using some fixed-point results on gauge spaces, some existence results of positive solutions are obtained.

The meridian curve of a wetting drop: a boundary value problem and its elliptic integrals solution

In this article we study the shape of free surfaces of a static fluid under gravity. We consider the meridian curve of a heavy liquid drop standing on a horizontal base: the main assumption concerns the liquid wetting capability, namely its contact angle well below $$\pi /2$$ . The nonlinear differential boundary problem is solved through the shooting method. Our treatment is self-consistent as holding all demonstrations of existence, uniqueness, and computability. We conclude providing the eigenvalues set to the radius and the meridian curve of the drop through elliptic integrals: such a new exact solution—see (3.9) and (3.10) —is enriching the literature on capillarity.

Closed form Solution to Nonlinear Equilibria and Oscillations of a Damped Eccentric Rotor

We study an eccentric and elastically damped rotor from both a statical and dynamical point of view. The system, whose genesis is in the re-loading mechanism of an automatic watch, behaves like a generalized physical pendulum with the addition of eccentricity and damping. The static analysis is performed by settling the statical equilibria and defining their nature, whose effective computation can be done numerically. The dynamical analysis leads to a nonlinear differential initial-value problem whose integration is carried out by means of Jacobi elliptic functions. It reveals that, starting from both positional and kinetic zero initial conditions, only periodical motions, see formulae (4.8) or (4.13), are allowed and all confined inside a potential well. Closed form expressions of the oscillation period have been obtained through complete elliptic integrals of the first kind. In such a way a further treatment is added to the non-rich collection of 1-D nonlinear oscillators suitable of closed form integration.

Energy-minimising parallel flows with prescribed vorticity distribution

This note concerns a nonlinear differential equation problem in which both the nonlinearity in the equation and its solution are determined by other constraints. The question under consideration arises from a study of two-dimensional steady parallel-flows of a perfect fluid governed by Euler's equations and a free-boundary condition, when the distribution of vorticity is arbitrary but prescribed.

Direct approach for solving nonlinear evolution and two-point boundary value problems

Time-delayed nonlinear evolution equations and boundary value problems have a wide range of applications in science and engineering. In this paper, we implement the differential transform method to solve the nonlinear delay differential equation and boundary value problems. Also, we present some numerical examples including time-delayed nonlinear Burgers equation to illustrate the validity and the great potential of the differential transform method. Numerical experiments demonstrate the use and computational efficiency of the method. This method can easily be applied to many nonlinear problems and is capable of reducing the size of computational work.

Partial Differential Equations

The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric problems, and fourth order equations in conformal geometry.

On the axisymmetric spreading of non-Newtonian power-law gravity currents of time-dependent volume: An experimental and theoretical investigation focused on the inference of rheological parameters

We study axisymmetric gravity currents consisting of a constant or time-dependent volume of a power-law viscous fluid propagating on a horizontal rigid plane below a fluid of lesser density. The intruding fluid is considered to have a pure Ostwald–DeWaele power-law constitutive equation. First, the conditions for buoyancy–viscous balance are examined, and the current rate of spreading is derived with a box-model. An existing self-similar solution to the nonlinear differential problem for the influx of a constant or time-variable volume of fluid is then described. Results from a number of experiments conducted in a 30° sector with shear thinning, Newtonian and shear thickening fluids, and with constant and increasing release rate, are presented and interpreted with the theoretical solution, obtaining globally a very satisfactory agreement. The rheological parameters of the fluid, derived with a best fit procedure, are compared to those measured independently with conventional rheometry. Confidence intervals are evaluated for both estimates of flow behavior and consistency indices. Results support the feasibility of controlled constant flux laboratory experiments with gravity currents in axisymmetric geometry to infer the rheology of power-law fluids, especially at very low shear rates and with shear thinning fluids.

Effect of variable permeability on the propagation of thin gravity currents in porous media

A new formulation is proposed to study the influence of deterministic heterogeneity on the propagation of thin two-dimensional gravity currents in a porous medium above a horizontal impervious boundary. Heterogeneity is conceptualized as a monotonic power-law variation of medium permeability transverse or parallel to the direction of propagation. Considering the injection of a constant or time-variable volume of fluid, the nonlinear differential problem admits a similarity solution which describes the shape and rate of propagation of the current. The bounds on parameters necessary to respect model assumptions are derived asymptotically and for finite time, to clarify the range of applicability of the proposed models. An application to the migration of a contaminant gravity current in the subsurface is then discussed, showing the impact of permeability variations on extension and shape of the intrusion.

On shear thinning fluid flow induced by continuous mass injection in porous media with variable conductivity

A new formulation is proposed to examine the propagation of the pressure disturbance induced by the injection of a time-variable mass of a weakly compressible shear thinning fluid in a porous domain with generalized geometry (plane, radial, or spherical). Medium heterogeneity along the flow direction is conceptualized as a monotonic power-law permeability variation. The resulting nonlinear differential problem admits a similarity solution in dimensionless form which provides the velocity of the pressure front and describes the pressure field within the domain as a function of geometry, fluid flow behavior index, injection rate, and exponent of the permeability variation. The problem has a closed-form solution for an instantaneous injection, generalizing earlier results for constant permeability. A parameter-dependent upper bound to the permeability increase in the flow direction needs to be imposed for the expression of the front velocity to retain its physical meaning. An example application to the radial injection of a remediation agent in a subsurface environment demonstrates the impact of permeability spatial variations and of their interplay with uncertainties in flow behavior index on model predictions.

Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian

In this article, we consider a nonlocal problem involving the p(x)-Laplacian of the type Applying a three critical points theorem from Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), pp. 249–258], we obtain the existence of two intervals of positive real parameters λ for which the above problem admits three weak solutions in , whose norms are uniformly bounded with respect to λ belonging to one of the two open intervals. In particular, we also prove some properties of p(x)-Kirchhoff–Laplace operator.

CONTROLLABILITY FOR NONLINEAR VARIATIONAL EVOLUTION INEQUALITIES

In this paper we investigate the approximate controllability for the following nonlinear functional differential control problem: <TEX>$$x^{\prime}(t)+Ax(t)+{\partial}{\phi}(x(t)){\ni}f(t,x(t))+h(t)$$</TEX> which is governed by the variational inequality problem with nonlinear terms.

Similarity solutions for spreading of a two-dimensional non-Newtonian gravity current in a porous layer

We consider the motion of shallow two-dimensional gravity currents of a purely viscous and relatively heavy power-law fluid of flow behavior index n in a uniform saturated porous layer above a horizontal impermeable boundary, driven by the release from a point source of a volume of fluid increasing with time like tα. The equation of motion for power-law fluids in porous media is a modified Darcy’s law taking into account the nonlinearity of the rheological equation. Coupling the flow law with the mass balance equation yields a nonlinear differential problem which admits a self-similar solution describing the shape of the current, which spreads like t(α+n )/(2+n), generalizing earlier results for Newtonian fluids. For the particular values α=0 and 2, closed-form solutions are derived; else, a numerical integration is required; the numerical scheme is tested against the analytical solutions. Two additional analytical approximations, valid for any α, are presented. The space-time development of the gravity current is discussed for different flow behavior indexes.

Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in[0,1]\\[3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<α≤3 is a real number, and \(D^{\alpha}_{0+}\) is the Caputo’s fractional derivative, and f:[0,1]×[0,+∞)→[0,+∞) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.

Partial Differential Equations

The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric problems, regularity of free boundaries, conformal invariance and the Willmore functional.

A Nonlinear GMRES Optimization Algorithm for Canonical Tensor Decomposition

A new algorithm is presented for computing a canonical rank-$R$ tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-$R$ tensor consists of the sum of $R$ rank-one tensors. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the stand-alone one-step process as a preconditioner. In particular, the nonlinear extension of GMRES we use that was proposed by Washio and Oosterlee in [Electron. Trans. Numer. Anal., 15 (2003), pp. 165--185] for nonlinear partial differential equation problems (which is itself related to other existing acceleration methods for nonlinear equation systems). In the third step, a line search is performed for globalization. The resulting nonlinear GMRES (N-GMRES) optimization algorithm is applied to dense and sparse tensor decomposition test problems. The numerical tests show that ALS accelerated by N-GMRES may significantly outperform stand-alone ALS when highly accurate stationary points are desired for difficult problems. Further comparison tests show that N-GMRES is competitive with the well-known nonlinear conjugate gradient method for the test problems considered and outperforms it in many cases. The proposed N-GMRES optimization algorithm is based on general concepts and may be applied to other nonlinear optimization problems.

Localisation of coastal trapped waves by longshore variations in bottom topography

Variations in shelf geometry mean that a coastal trapped wave mode can propagate within some finite length of shelf but be evanescent outside this region. This paper constructs such geographically localised coastal trapped waves using a WKBJ approximation. Comparison with full numerical solutions of the non-linear differential eigenvalue problem demonstrates that the approximation is extremely accurate. The asymptotic and full numerical models are then used to examine the parameters and geometries that govern the existence of these modes.

SOME MULTIPLICITY RESULTS TO THE EXISTENCE OF THREE SOLUTIONS FOR A DIRICHLET BOUNDARY VALUE PROBLEM INVOLVING THE P-LAPLACIAN

In this paper we prove the existence of two intervals of positive real parameters λ for a Dirichlet boundary value problem involving the p-Laplacian which admit three weak solutions, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. Our main tool is a three critical points theorem due to G. Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim., 28:249–258, 2004].