A finite element method for the general solution of ordinary differential equations

A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.     

A seventh-order numerical method for solving boundary value nonlinear ordinary differential equations

A numerical method for solving two-point boundary value problems associated with systems of first-order nonlinear ordinary differential equations is described. It needs three function evaluations for each sub-interval and is of order O(h⁷), where h is the space chop. Results of computational experiments comparing this method with other known methods are given.     

A cluster series expansion technique for the spectral solution of differential equations

 A cluster series expansion technique for the spectral solution of differential equations is presented. An alternating fixed-free subspace strategy used for improving the accuracy of the cluster series expansion generates a family of m-step iterative algorithms solving large dense systems of linear equations of order m×n as a few partial systems of order n+m−1. The computational behaviour of the proposed m-step algorithms is examined by solving dense systems of linear equations up to the order 40 000 using the m-step algorithms for m = 2, 4, 6, 8, and 10.     

A numerical method for solving two-point boundary value problems for nonlinear ordinary differential equations

A numerical method for solving two-point boundary value problems associated with systems of first-order nonlinear ordinary differential equations is described. It needs four function evaluations at each point and is of order h⁶, where h is the space chop. Results of computational experiments, which include perturbation of the initial conditions, comparing this method with other known methods are given.     

A homotopy analysis method for the nonlinear partial differential equations arising in engineering

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.     

Traveling wave solutions of nonlinear partial differential equations

Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Bäcklund Transformation.     

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.     

A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $L_2,L_{\infty }$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).     

Two high-order numerical methods for solving boundary-value nonlinear ordinary differential equations

Two numerical methods for solving two-point boundary-value problems associated with systems of first-order nonlinear ordinary differential equations are described. The first method, which is based on Lobatto quadrature, requires four internal function evaluations for each subinterval. It does not need derivatives and is of order h⁷, where h is the space chop. The second method, which is similar to the first but is based on Lobatto–Hermite quadrature, makes the additional use of derivatives to achieve O(h⁹) accuracy. Results of computational experiments comparing these methods with other known methods are given.     

Piecewise polynomials and the partition method for nonlinear ordinary differential equations

Summary An efficient numerical method, used previously for linear differential equations [1], is here extended to systems of nonlinear ordinary differential equations. Spline functions are used as the basic approximations. Residuals are liquidated by setting their integrals equal to zero over specified subintervals of the intervals of analyticity. Several diverse examples are given.Notation a Radius of circular membrane - [C _j ] An unsymmetrical 4×4 matrix, defined by Eq. (6) - [C _j ] An unsymmetrical 6×6 matrix, defined by Eq. (8) - E Young's modulus - F _i Algebraic or transcendental functions - L Differential operator - R Closed interval - T Dimensionless radius stress for the circular membrane - [Y _j ] The transpose of the row matrix [y _j –1,y _j –1,y _j ,y _j ] - [Y _j ] The transpose of the row matrix [y _j –1,y _j –1,y _j–1 ,y _j ,y _j ,y _j ] - Residual - Poisson's ratio Formerly with Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, Illinois, USA.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

Optical solitons of some fractional differential equations in nonlinear optics

This paper deals with two different forms of the conformable fractional Benjamin–Bona–Mahony (BBM) equations by an analytical method. These physical models have important applications for describing the propagation of optical pulses in non-linear media. The conformable fractional symmetric BBM equation and the conformable time fractional Equal-width (EW) equation are considered. The extended Jacobi’s elliptic function expansion scheme are used to extract explicit solitons.     

A numerical method for coupled differential equations

This paper describes a numerical method for solving first-order coupled matrix differential equations. Recursive equations are used to find ‘reflection’, ‘transmission’ and ‘source’ coefficients; these coefficients are then used to construct the vector solution to the differential equations. The method can be used to solve linear and non-linear differential equations with specified initial or two-point boundary values. Numerical results for several initial value problems are given in the paper.     

Impact force identification using the general inverse technique

The general inverse technique has been applied to the problem of impact force identification. The force identification problem can be stated as follows: ‘Given the response of a structure, what impact force is required to produce this response?’ Experimental measurement data was used to describe the velocity response of an impacted cantilever beam. The inverse technique used these measurements to predict the impact force which resulted in the beam response.

A spectrum analyzer was used to measure the beam accelerations within 40 μs time intervals. The accelerations were integrated so that the response was in suitable velocity form for use in the inverse algorithm. The analyzer and impact hammer also allowed the measurement of the actual dynamic load, and thus provided a means of comparison against the predicted impact force given by the inverse technique.     

Solution of nonlinear partial differential equations using the Chebyshev spectral method

The spectral method is a powerful numerical technique for solving engineering differential equations. The method is a specialization of the method of weighted residuals. Trial functions that are easily and exactly differentiable are used. Often the functions used also satisfy an orthogonality equation, which can improve the efficiency of the approximation. Generally, the entire domain is modeled, but multiple sub-domains may be used. A Chebyshev-Collocation Spectral method is used to solve a two-dimensional, highly nonlinear, two parameter Bratu's equation. This equation previously assumed to have only symmetric solutions are shown to have regions where solutions that are non-symmetric in x and y are valid. Away from these regions an accurate and efficient technique for tracking the equation's multi-valued solutions was developed.It is found that the accuracy of the present method is very good, with a significant improvement in computer time.     

Automatic identification of differential equations simulating the behavior of neuron circuits

A method is described for computing the Markov parameters of a matrix from the relevant parameters, and for constructing the entire differential equation sought. Evaluations are made of the convergence of the proposed methods of approximation, and a demonstration is given of the practical application of the theory to the identification of respiratory neuron circuits. Translated from Izmeritel'naya Tekhnika, No. 3, pp. 66–68, March, 1994.     

A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects

A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.