A CLASS OF BLOCK MULTISTEP METHODS FOR SOLVING GENERAL THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS

The numerical solutions of general third order initial value problems of ordinary di�erential equations have been studied in this research work. A new class of block multistep methods capable of solving general third order IVPs of ODEs using variable step size technique have been developed. Collocation and interpolation of power series as the approximate solution is adopted. The block multistep method was intensi�ed by the introduction of continuous scheme in order to circumvent the limitation created by reducing to systems of �rst order ODEs. The new class of variable step-size method has the advantage to control and minimize error, determine and vary the step size as well as decide the prescribed tolerance level to ascertain the maximum errors. Some theoretical properties of the block multistep methods such as order of the scheme, zero stability, consistency and determination of the region of absolute stability of the scheme have been conducted and presented. Numerical examples on nonsti� IVPs have been used to test the performance of the methods, in addition, comparing the maximum error as the prescribed tolerance parameter level is reduced in the method. The newly developed methods have been written as mathematical program and expressed in form of mathematical language which can run simultaneously when implemented. The newly formulated variable step-size block multistep methods perform better when compared with other existing methods as the prescribed tolerance parameter level got smaller and smaller. Furthermore, the newly developed methods possess the attribute to control and decide on the estimate of the actual step size that will guarantee an improved results with better maximum errors. This, in particular, is seen as an advantage of the variable step size method over other existing methods approximated with �xed step size. Finally, the idea of predictor-corrector methods used by various researchers to predict and correct estimates has been extended in the newly proposed method to change/decide on suitable step size, determine the prescribed tolerance level and error control/minimization

Four Steps Implicit Method for the Solution of General Second Order Ordinary Differential Equations

Four steps implicit scheme for the solution of second order ordinary differential equation was derived through interpolation and collocation method. Newton polynomial approximation method was used to generate the unknown parameters in the corrector. The method was tested with numerical examples and it was found to be efficient in solving second order ordinary differential equations

Common fixed point theorems for non-self mappings of nonlinear contractive maps in convex metric spaces

In this paper, we introduce a class of nonlinear contractive mappings in metric space. We also establish common fixed point theorems for these pair of non-self mappings satisfying the new contractive conditions in the convex metric space . An example is given to validate our results. The results generalize and extend some results in literature

K-Step Block Predictor-Corrector Methods for Solving First Order Ordinary Differential Equations

A K-step block Predictor-Corrector Methods for solving first order ordinary differential equations are formulated and applied on non-stiff and mildly stiff problems using variable step size technique. In this method, collocation and interpolation of the power series as the approximate solution is carried out with aim of generating the continuous scheme. The investigation of some selected theoretical properties of the method is analysed as well as determination of the region of absolute stability of the method. In addition, the implementation of the proposed method is done by applying variable step size techniqu

A 5-Step Block Predictor and 4-Step Corrector Methods for Solving General Second Order Ordinary Differential Equations

A 5-step block predictor and 4-step corrector methods aimed at solving general second order ordinary differential equations directly will be constructed and implemented on non-stiff problems. This method, which extends the work of block predictor-corrector methods using variable step size technique possess some computational advantages of choosing a suitable step size, deciding the stopping criteria and error control. In addition, some selected theoretical properties of the method will be investigated as well as determination of the region of absolute stability. Numerical results will be given to show the efficiency of the new metho

Programming codes of block-Milne's device for solving fourth-order ODEs

Block-Milne’s device is an extension of block-predictor-corrector method and specifically developed to design a worthy step size, resolve the convergence criteria and maximize error. In this study, programming codes of block- Milne’s device (P-CB-MD) for solving fourth order ODEs are considered. Collocation and interpolation with power series as the basic solution are used to devise P-CB-MD. Analysing the P-CB-MD will give rise to the principal local truncation error (PLTE) after determining the order. The P-CB-MD for solving fourth order ODEs is written using Mathematica which can be utilized to evaluate and produce the mathematical results. The P-CB-MD is very useful to demonstrate speed, efficiency and accuracy compare to manual computation applied. Some selected problems were solved and compared with existing methods. This was made realizable with the support of the named computational benefit

Classes of Ordinary Differential Equations Obtained for the Probability Functions of Gumbel Distribution

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODEs) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the Gumbel distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are new ways of understanding the nature of the probability functions that characterize the distribution. The method can be extended to other probability distributions, functions and can serve an alternative to approximation and estimation

Block Algorithm for General Third Order Ordinary Differential Equation

We present a block algorithm for the general solution o

EXISTENCE, UNIQUENESS AND STABILITY OF A MILD SOLUTION OF LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

We introduce the concept of a mild solution of Lipschitzian quantum stochastic differential equations (QSDEs). Results on the existence, uniqueness and stability of a mild solution of QSDEs are established. This is accomplished within the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus. Here, the results on a mild solution are weaker compared with the ones in the literature

Solution of Differential Equations by Three Semi-Analytical Techniques

In this work, we present some semi-analytical techniques namely Differential Transform Method (DTM), Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM) for the solution of differential equations. The equations considered include initial value problems and boundary value problems. The results indicated that DTM is easy to apply but requires transformation, while ADM does not need any transformation except the calculation of Adomian polynomials. In addition, it was demonstrated that HPM involves perturbation and more computations. The results obtained converged rapidly to the exact solution