Formulating Mathematica pseudocodes of block-Milne's device for accomplishing third-order ODEs (original) (raw)
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Programming codes of block-Milne's device for solving fourth-order ODEs
International Journal of Advanced and Applied Sciences, 2018
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 benefits.
Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code
Journal of Mathematics and Statistics, 2012
Problem statement: This study is concerned with the development of a code based on 2point block method for solving higher order Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) directly. Approach: The block method was developed based on numerical integration and using interpolation approach which is similarly as Adams Moulton type. Furthermore, the proposed method is derived in order to solve higher order ODEs in a single code using variable step size and implemented in a predictor corrector mode. This block method will act as simultaneous numerical integrator by computing the numerical solution at two steps simultaneously. Results: The numerical results for the direct block method were superior compared to the existing block method. Conclusion: It is clearly proved that the code is able to produce good results for solving higher order ODEs.
Block methods as an approach for solving higher order ordinary differential equations (ODEs) have been seen to be very useful in recent literature. However, the development of the block methods for higher order, such as fourth order ODEs is seen to include a lot of steps and transformations. This is irrespective of the approach adopted; be it interpolation, numerical integration or Taylor series. Hence, this study investigates into producing an algorithm that can produce the desired block method directly for any value of the stepnumber k, whose computational complexity is also shown.
Step Block Method Algorithm for Arbitrary Order Ordinary Differential Equations
2016
Block methods have been very suitable and useful in the solution of higher order ordinary differential equations (ODEs). This is because it gives a higher level of accuracy than conventional methods of reduction of higher order ODEs or predictor-corrector methods. However, the step by step process needed in developing this method based on either the stepnumber or the order of the ODE under consideration is rigorous. Hence, the introduction of an algorithm that will bypass this setback is expedient and thus the aim of this article. Mathematics Subject Classifications: 65L05, 65L06, 65L10
k−Step Block Method Algorithm for Arbitrary Order Ordinary Differential Equations
Block methods have been very suitable and useful in the solution of higher order ordinary differential equations (ODEs). This is because it gives a higher level of accuracy than conventional methods of reduction of higher order ODEs or predictor-corrector methods. However, the step by step process needed in developing this method based on either the stepnumber or the order of the ODE under consideration is rigorous. Hence, the introduction of an algorithm that will bypass this setback is expedient and thus the aim of this article.
Mathematica Pseudocodes for Implementing Block Adams Family
Authorea, 2020
This study is considered to formulate Mathematica pseudocodes for implementing block Adams family (MPIBAF). An idea multinomial basis function approximant will be utilized to process the interpolation and collocation methods. A special block Adams family in form of block Adams-Bashforth and Block Adams-Moulton methods will be developed via interpolation and collocation method to foster the principal local truncation error thereby bringing into existence the convergency limits. The application of Mathematica pseudocodes will be processed on some applied math problems in a parallel manner with each processor depending on one another. The computable results will be produced via a compiled Mathematica pseudocode in a gradual manner. In addition, technical computation supersedes manual computation as demonstrated by the results of the numerous gains such as ease of computable processes and structures, better accuracy and quicker convergency.
New computational method for solving ordinary differential equations
2011
In this paper we present a developed couple block method for solving first order ordinary differential equations (ODEs). The coupled block method consists of two proposed block methods i.e the two point two step block method of order five and three point two step block method of order six. Therefore, these methods will estimate the numerical solutions at two and three points simultaneously within a block. The proposed block method is derived using Lagrange interpolation polynomial and is presented as in the simple form of the Adams Moulton type. The developed code is implemented using variable step size and order. The stability of the methods is also studied. Numerical results are presented to compare the performance of the developed code to the existence block method.
2018
The idea of technological computing has immensely assisted to enhance accuracy and maximize computed errors involving computational math. Softcodes computer programme is guided towards supplying comfortable computation, proficiency and faster results at all times. The objective of this study will be to devise softcodes of parallel processing Milne’s device (SPPMD) via exponentially fitted method for valuating special ordinary differential equations. This is established through collocation and interpolation of the exponentially fitted method. Dissecting (SPPMD) produces the principal local truncation error (PLTE) after expressing the order of SPPMD leading to the boundary of convergence. Some selected examples of special ODEs were tested to show the efficiency and accuracy of (SPPMD) at different boundary of convergence. The finished results exist with the aid of (SPPMD). Computed results show that the (SPPMD) is more proficient compare to subsisting methods in terms of the work out ...
Order Six Block Integrator for the Solution of First-Order Ordinary Differential Equations.
In this research work, we present the derivation and implementation of an order six block integrator for the solution of first-order ordinary differential equations using interpolation and collocation procedures. The approximate solution used in this work is a combination of power series and exponential function. We further investigate the properties of the block integrator and found it to be zero-stable, consistent and convergent. The block integrator is further tested on some real-life numerical problems and found to be computationally reliable. Block integrators for solving ODEs have been proposed by W. E. Milne[18] who used them as starting values for predictor-corrector algorithm, [8] developed Milne's method in the form of implicit integrators and [12] also contributed greatly to the development and application of block integrators. Various authors [1], [3], [4], [5], [9], [11], [17] and [19] proposed LMMs to generate numerical solution to (1). They proposed integrators in which the approximate solution ranges from power series, Chebychev's, Lagrange's and Laguerre's polynomials. The advantages of LMMs over single step methods have been extensively discussed in [2].