Oscillatory property of higher order nonlinear difference equations

In this paper, we study the oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form

We obtain some necessary and sufficient conditions for all bounded solutions of (*) to be oscillatory and for (*) to have a nonoscillatory solution of a special form.     

A higher order method for determining nonisolated solutions of a system of nonlinear equations

In this note, we obtain a method of order at least four to solve a singular system of nonlinear algebraic equations. This is achieved by enlarging the system to a higher dimensional one whose solution is isolated. For the larger system we use a method developed by B. Neta.     

A higher order finite element algorithm for the unsteady Navier-Stokes equations

A higher order element, the Tocher 10 or C₀ Cubic on triangles, is the base for formulation of a finite element algorithm for numerical calculation of fluid flows governed by the unsteady Navier-Stokes equations. Results from the calculation of supersonic free shear layer flow are numerically accurate and in excellent agreement with finite difference solutions. Diverse characteristics for these two classes of methods emerge when the requirements of core storage and computer time are considered.     

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.     

A Schur vector method to solve higher order Lyapunov equations

A general Schur vector method for solving a partial differential equation which appears in designing nonlinear optimal control laws for linear or nonlinear dynamical systems is proposed. The method reduces the dimensionality problem of the direct method. Numerical experiments show that it also reduces the numerical instability of the eigenvector method     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

A new method based on legendre polynomials for solution of system of fractional order partial differential equations

We study legendre polynomials in case of more than one variable and develop new operational matrices of fractional order integrations as well as fractional order differentiations. Based on these operational matrices, we develop a new sophisticated technique to solve a coupled system of fractional order partial differential equations. Our technique reduces the coupled system under consideration to a system of easily solvable algebraic equations without discretizing the system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique matches well with the exact solutions of the problems. We also study error analysis graphically.     

The finite difference method for first order impulsive partial differential-functional equations

We consider initial boundary value problems for first order impulsive partial differential-functional equations. We give sufficient conditions for the convergence of a general class of one step difference methods. We assume that given functions satisfy the non-linear estimates of the Perron type with respect to the functional argument. The proof of stability is based on a theorem on difference functional inequalities generated by an impulsive differential-functional problem. It is an essential assumption in our consideration that given functions satisfy the Volterra condition. We give a numerical example.     

An observer-canonical state-representation for a system of linear variable difference equations

This short paper presents a technique to convert a well-formed system of linear variable difference equations to a corresponding system of state equations. The technique applies to both causal and noncausal systems. The resulting state-representation is in the observer version of the Luenberger canonical form.     

A generalized higher order neural network for aircraft recognition in a video docking system

In modern world of today where air traffic is continuously increasing and available space at the airports remains finite, there is a problem of safe docking of aircraft. The problem needs to be solved to ensure safe and smooth movement of aircraft, passengers and crew while making optimum utilization of available ground space. Without such systems having in place, accidents keep occurring due to human judgment errors. These accidents are causing loss of material costs and human injury. The importance of Video-based Docking Systems is continuously increasing due to the challenges of current and upcoming traffic demands of future. This paper evaluates two neural networks architectures for recognition of civil airliners in a Video docking system. The networks compared are feedforward neural network and probabilistic neural network. The results are compared by presenting data to neural networks while deforming the shape, adding noise and partly occluding the shape and presenting those angles for which network was not trained.     

The Vallée-Poussin problem for higher order nonlinear hyperbolic equations

For higher order nonlinear hyperbolic equations nonclassical boundary value problems of Vallée-Poussin type are studied, and unimprovable in a sense sufficient conditions of solvability, unique solvability and well-posedness are established.     

Exp-function method for solving nonlinear evolution equations with higher order nonlinearity

In this paper, the Exp-function method is used to obtain generalized solitary solutions of the generalized Drinfel’d–Sokolov–Wilson (DSW) system and the generalized (2+1)-dimensional Burgers-type equation. Then, some of the solitary solutions are converted to periodic solutions or hyperbolic function solutions by a simple transformation. The results show that the Exp-function method is a powerful and convenient mathematical tool for solving nonlinear evolution equations with higher order nonlinearity.     

CP methods of higher order for Sturm-Liouville and Schrödinger equations

The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259-277, is based, is extended to higher order. The implementation of the original algorithm, which was of order {12,10} (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders {14,12}, {16,14} and {18,16} are introduced. Besides regular Sturm-Liouville and one-dimensional Schrödinger problems also radial Schrödinger equations are considered with potentials of the form V(r)=S(r)/r+R(r), where S(r) and R(r) are well behaved functions which tend to some (not necessarily equal) constants when r→0 and r→∞. Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.     

A new cubic convergent method for solving a system of nonlinear equations

We present a new cubic convergent method for solving a system of nonlinear equations. The new method can be viewed as a modified Chebyshev's method in which the difference of Jacobian matrixes replaces three order tensor. Therefore, the new method reduces the storage and computational cost. The new method possesses the local cubic convergence as well as Chebyshev's method. A rule is deduced to ensure the descent property of the search direction, and a nonmonotone line search technique is used to guarantee the global convergence. Numerical results indicate that the new method is competitive and efficient for some classical test problems.     

A new class of interval methods with higher order of convergence

In this paper we introduce a new class of interval methods for enclosing a simple root of a nonlinear equation. For each nonnegative integerp we describe an iterative procedure belonging to this class which requiresp+1 function values and an interval evaluation of the second derivative per step. The order of convergence of the iterative procedure grows exponentially withp. Forp≥4 this order is strictly greater than \(\left( {\frac{{1 + \sqrt 5 }}{2}} \right)^{p + 2} \) .