Order Nonlinear Partial Differential Equations Research Articles

Special solutions of a high-order equation for waves in a liquid with gas bubbles

A fifth--order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling waves solution are constructed. Connection of self--similar solutions with Painleve transcendents and their high--order analogous is discussed.

A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

Optimal advertising and pricing in a class of general new-product adoption models

In [21], Sethi et al. introduced a particular new-product adoption model. They determine optimal advertising and pricing policies of an associated deterministic infinite horizon discounted control problem. Their analysis is based on the fact that the corresponding Hamilton–Jacobi–Bellman (HJB) equation is an ordinary non-linear differential equation which has an analytical solution. In this paper, generalizations of their model are considered. We take arbitrary adoption and saturation effects into account, and solve finite and infinite horizon discounted variations of associated control problems. If the horizon is finite, the HJB-equation is a 1st order non-linear partial differential equation with specific boundary conditions. For a fairly general class of models we show that these partial differential equations have analytical solutions. Explicit formulas of the value function and the optimal policies are derived. The controlled Bass model with isoelastic demand is a special example of the class of controlled adoption models to be examined and will be analyzed in some detail.

A modification of He's variational iteration method by Taylor's series for solving second order nonlinear partial differential equations

In this work, a modification of He's variational iteration method by Taylor's series is used for finding the solution of the second order nonlinear partial differential equations. This modification expands the application of variational iteration method for those nonlinear equations which have logarithmic or trigonometric nonlinear part. To show the efficiency of the method, several examples are presented.

The alternating evolution methods for first order nonlinear partial differential equations

In this paper, we present a brief survey of the recent developments in alternating evolution (AE) methods for numerical computation of first order partial differential equations, with hyperbolic conservation laws and Hamilton-Jacobi equations as two canonical examples. The main difficulty of such computation arises from the nonlinearity of the model, making it necessary to incorporate an appropriate amount of numerical viscosity to capture the entropy/viscosity solution as physically relevant solutions. The alternating evolution method is based on the AE system of the original PDEs, the discretization technique ranges from finite difference, finite volume and the discontinuous Galerkin methods. In all these cases, the AE solver can produce accurate solutions with equal computational time than the traditional solvers. In particular, the AE formulation allows the same discontinuous Galerkin discretization for both conservative and non-conservative PDEs under consideration. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use simplified models to describe the details of the AE method. Sample simulation results on a few models are also given.

On the solution of confined aquifer flow equations: finite difference approximations

In general, groundwater flow in porous media on a microscopic level is three-dimensional. However, when considering regional flow problems i.e. groundwater flow on a macroscopic level, it is noted that because of the low ratio of aquifer thickness to horizontal length, the flow in the aquifer is practically horizontal[21]. Flow in confined aquifers is therefore described by a two-dimensional second order non-linear partial differential equation with variable coefficients on a finite domain. In this paper, we examine the case of a confined aquifer that is being pumped

The upper–lower solution method for the coupled system of first order nonlinear PDEs

This paper is concerned with a coupled system of first order nonlinear partial differential equations. This system is, but not limited in, the extended case of the general blood–tissue exchange model (BTEX). We use the solutions of a coupled system of first order ordinary differential equations to construct a pair of ordered lower and upper solutions for the nonlinear partial differential system. By monotone iterative methods we show the existence and uniqueness of the solution of the coupled system of nonlinear partial differential equations. The asymptotic behavior of the solutions to the coupled nonlinear partial differential system can be obtained by investing the asymptotic behavior of the solution to the coupled system of first order ordinary differential equations. Finally we apply these results to the mathematical models of general blood–tissue exchange and the gas–solid inter-phase heat transfer for the fast igniting catalytic converter problems.

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by com- putational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.

Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a Singular Nonlinearity Modeling a MEMS Capacitor

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric 2D case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at (t,x)=(0,0)∈C 2 . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the k-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

Homotopy method for a mean curvature-based denoising model

Variational image denoising models based on regularization of gradients have been extensively studied. The total variation model by Rudin, Osher, and Fatemi (1992) [38] can preserve edges well but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To overcome this, mean curvature-based energy minimization models offer one approach for restoring both smooth (no edges) and nonsmooth (with edges) images. As such models lead to fourth order (instead of the usual second order) nonlinear partial differential equations, development of fast solvers is a challenging task. Previously stabilized fixed point methods and their associated multigrid methods were developed but the underlying operators must be regularized by a relatively large parameter. In this paper, we first present a fixed point curvature method for solving such equations and then propose a homotopy approach for varying the regularized parameter so that the Newton type method becomes applicable in a predictor–corrector framework. Numerical experiments show that both of our methods are able to maintain all important information in the image, and at the same time to filter out noise.

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.

Soliton Equations and Higher Order Nonlinear Partial Differential Equations

Some nonlinear PDEs partial differential equations are exactly solvable. As an example, nonlinear PDEs such as the Soliton equation were shown to be exactly solvable by quantum operator methods. More recently the Boltzmann equation was solved exactly as well. A question is are other equations such as general arbitrary order nonlinear PDEs solvable. Also recent generalizations of nonextensive statistics has introduced a nonlinear in power of the distribution PDE equation, which are solved for linear drift coefficients by the power-law distribution derived from the Tsallis nonextensive statistics and for which we have recently presented an exact solution for arbitrary nonlinear drift coefficients [3]. Are there possible general solution methods for nonlinear partial differential equations. One possible approach that we present in this letter depends on the ability to transform generally nonlinear PDEs of arbitrary order to 2nd order standard form Fokker-Planck PDEs which have been shown to have exact short time transition probability solutions irregardless of the nonlinear form of the drift and diffusion coefficients. We discuss these questions briefly in the following derivation of a solution to the KdV type of third order nonlinear PDE.

Stochastic ranking process with time dependent intensities

We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the infinite particle limit. We give an explicit formula for the limit distribution and show that the limit distribution function is a unique global classical solution to an initial value problem for a system of a first order non-linear partial differential equations with time dependent coefficients.

A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. A high order nonlinear partial differential equation (PDE) needs to be solved, and the gradient descent method usually takes high computational cost. In this paper, we propose a fast and efficient numerical algorithm to solve minimization problems related to Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

Existence of Classical Solutions near Characteristic Points of First Order Nonlinear Partial Differential Equations

We treat a nonlinear partial differential equation F(x,u,ux) = 0 in a neighborhood of x = 0 ∈ Rd, where F(x,u,p) is a real-valued smooth function. It is well-known that solutions are constructed by solving noncharacteristic Cauchy problem with the method of characteristics, provided Fpi(0,0,0) ≠ 0 for some i. In this paper we study the existence of a classical solution under the condition that Fpi(0,0,0) = 0 for all 1 ≤ i ≤ d.

Size-structured populations: immigration, (bi)stability and the net growth rate

We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalised net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero, i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by Allée-effect.

Oscillation results for higher even order nonlinear partial functional differential equations of neutral type

For some higher even order nonlinear partial functional differential equations of neutral type with continuous distributed delay, we obtain new oscillation criteria of solutions for boundary value problem of the equations, in our paper we omit the assumption which has been required for related results given before. The main tool used is the standard Philos integral averaging technique.

Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.