The spline alternating group explicit (splage) method for two dimensional problems

A new group explicit iterative method based on cubic spline approximations is presented for the numerical solution of partial differential equations. The numerical results obtained confirm the viability of the method.     

Optimization free neural network approach for solving ordinary and partial differential equations

Current work introduces a fast converging neural network-based approach for solution of ordinary and partial differential equations. Proposed technique eliminates the need of time-consuming optimization procedure for training of neural network. Rather, it uses the extreme learning machine algorithm for calculating the neural network parameters so as to make it satisfy the differential equation and associated boundary conditions. Various ordinary and partial differential equations are treated using this technique, and accuracy and convergence aspects of the procedure are discussed.

     

Sextic spline solution of variable coefficient fourth-order parabolic equations

We report new three-level implicit methods of O(k ²+h ⁴) and O(k ⁴+h ⁴) for the numerical solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficients. We use sextic spline in space and finite difference in time directions. Sextic spline function relations are derived by using off-step points. The linear stability of the presented method is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existence methods shows the superiority of our presented scheme.     

Stability of 2h-step spline method for delay differential equations

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.     

Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution

Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results.     

A B-spline collocation method for solving the incompressible Navier-Stokes equations using an ad hoc method: the Boundary Residual method

Collocation methods using piece-wise polynomials, including B-splines, have been developed to find approximate solutions to both ordinary and partial differential equations. Such methods are elegant in their simplicity and efficient in their application. The spline collocation method is typically more efficient than traditional Galerkin finite element methods, which are used to solve the equations of fluid dynamics. The collocation method avoids integration. Exact formulae are available to find derivatives on spline curves and surfaces. The primary objective of the present work is to determine the requirements for the successful application of B-spline collocation to solve the coupled, steady, 2D, incompressible Navier-Stokes and continuity equations for laminar flow. The successful application of B-spline collocation included the development of ad hoc method dubbed the Boundary Residual method to deal with the presence of the pressure terms in the Navier-Stokes equations. Historically, other ad hoc methods have been developed to solve the incompressible Navier-Stokes equations, including the artificial compressibility, pressure correction and penalty methods. Convergence studies show that the ad hoc Boundary Residual method is convergent toward an exact (manufactured) solution for the 2D, steady, incompressible Navier-Stokes and continuity equations. C¹ cubic and quartic B-spline schemes employing orthogonal collocation and C² cubic and C³ quartic B-spline schemes with collocation at the Greville points are investigated. The C³ quartic Greville scheme is shown to be the most efficient scheme for a given accuracy, even though the C¹ quartic orthogonal scheme is the most accurate for a given partition. Two solution approaches are employed, including a globally-convergent zero-finding Newton's method using an LU decomposition direct solver and the variable-metric minimization method using BFGS update.     

The direct solution of banded linear systems by a new matrix factorization procedure

In the finite difference/element discretisation of boundary value problems involving partial differential equations there occurs the problem of solving large order systems of banded linear systems.

Previously, direct methods of solution considered were the usual Gaussian elimination and LU decomposition strategies. In this paper an alternative factorization strategy is presented.     

Approssimazione bilaterale A-stabile nella integrazione di equazioni differenzali ordinarie

Following a previous paper [1] we estabilish conditions for the bilateral approximation of the solution of ordinary differential equations usingA-stable predictor-corrector formulas. An extension to parametric formulas is carried out and numerical examples concerned with stiff equations are presented.

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Dedicato al Professor S. Faedo in occasione del suo settantesimo compleanno     

Spline functions in chemistry: approximation of surfaces over triangle domains

Chemists often come across triangle domains – usually with the basic simplex in ?². A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically.

Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour – Liquid equilibrium data and explosion-limit surfaces of ternary gas systems).

A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed.     

A Bivariate Spline Method for Second Order Elliptic Equations in Non-divergence Form

A bivariate spline method is developed to numerically solve second order elliptic partial differential equations in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya–Babuska–Brezzi condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. Computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.     

Comparison of collocation methods for the solution of second order non-linear boundary value problems

This article concerns the application of cubic spline collocation tau-method for solving non-linear second order ordinary differential equations. Three collocation methods [Taiwo, O.A., 1986, A computational method for ordinary differential equations and error estimation. MSc dissertation, University of Ilorin, Nigeria (unpublished); Taiwo, O.A., 2002, Exponential fitting for the solution of two point boundary value problem with cubic spline collocation tau-method. International Journal of Computer Mathematics, 79(3), 229–306.] are discussed and applied to some second order non-linear problems. They are standard collocation, perturbed collocation, and exponentially fitted collocation. Numerical examples are given to illustrate the accuracy, efficiency and computational cost.     

A novel meshless collocation solver for solving multi-term variable-order time fractional PDEs

This paper presents a novel meshless collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, it employs the Fourier series expansion for spatial discretization, which transforms the original multi-term VOTFPDEs into a sequence of multi-term variable-order time fractional ordinary differential equations (VOTFODEs). Then, these VOTFODEs can be solved using the recent-developed backward substitution method. Several numerical examples verify the accuracy and efficiency of the proposed numerical approach in the solution of multi-term VOTFPDEs.

     

Solving parabolic PDE's by a decomposition method

The paper deals with the application of the hybrid method of decomposition for the solution of two-dimensional parabolic partial differential equations with constant coefficients. The convergence and accuracy of the CSDT method are analysed, deriving the form of the relationship between ‘eigenvalues’ and the time step. Theoretical problems of this method of decomposition are mentioned. Also, the expression for computing the coefficients for various types of boundary problems has been derived. Practical applications of the method are presented. The cubic spline function interpolation is used for the continuous signal reproduction in the hybrid system. Some results obtained are presented at the end.     

High Accuracy Difference Formulae For A Fourth Order Quasi-Linear Parabolic Initial Boundary Value Problem Of First Kind

In this paper, new three level implicit finite difference methods of O(k^2+h^2) and O(k^2+h^4) are proposed for the numerical solution of fourth order quasi-linear parabolic partial differential equations in one space variable, where k\gt 0 and h\gt 0 are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \partial u/\partial x is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.     

On the stability of a class of non-linear continuous systems†

A procedure to find the existence and domain of asymptotic stability for a class of systems described by a set of non-linear partial differential equations is presented in the context of semigroup theory.

No a priori bound is imposed on the non-linearity which is assumed to be a quad ratic (bilinear) or multilinear form.

The linearized system can be non-self-adjoint and need not be asymptotically stable equation by equation.

The solution of the stability problem is carried out. in two steps :

(a) the development of an L₂-like norm that bounds above the sup norm, a requirement to obtain a useful estimate of the non-linear term, and

(a)the computation of an estimate of the L₂-likc norm of the semigroup generated by the linearized system in terms of its usual L₂ norm.     

Systolic Preconditioned Algorithms for the Iterative Solution of Linear Systems

Systolic algorithms for preconditioned iterative procedures are described and discussed in relation to existing systolic arrays for iterative methods applied to the systems of linear equations arising from the solution of partial differential equations. In boundary value problems it is shown that if the cost of the preconditioned systolic arrays in terms of hardware is related to the (standard) iterative methods, then savings in the number of array cells can be made when the system involved is large and sparse (narrow bandwidth) with a significant improvement in convergence rate.     

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

MHD flow of radiative micropolar nanofluid in a porous channel: optimal and numerical solutions

The flow of a radiative and electrically conducting micropolar nanofluid inside a porous channel is investigated. After implementing the similarity transformations, the partial differential equations representing the radiative flow are reduced to a system of ordinary differential equations. The subsequent equations are solved by making use of a well-known analytical method called homotopy analysis method (HAM). The expressions concerning the velocity, microrotation, temperature, and nanoparticle concentration profiles are obtained. The radiation tends to drop the temperature profile for the fluid. The formulation for local Nusselt and Sherwood numbers is also presented. Tabular and graphical results highlighting the effects of different physical parameters are presented. Rate of heat transfer at the lower wall is seen to be increasing with higher values of the radiation parameter while a drop in heat transfer rate at the upper wall is observed. Same problem has been solved by implementing the numerical procedure called the Runge–Kutta method. A comparison between the HAM, numerical and already existing results has also been made.

     

A computational solution for the matrix riccati equation using laplace transforms

The solution for the finite-time matrix Riccati equation is presented in this paper. The solution to the Riccati equation is obtained in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and solved using Laplace transforms and the computation is done by the digital computer.

A numerical example for the proposed method is given.