Softcodes of Parallel Processing Milne’s Device via Exponentially Fitted Method for Valuating Special ODEs (original) (raw)
Related papers
2018
Over the years, scientific computing has contributed immensely to computational mathematics. Mathematica computer programming codes is known to provide computation and quick results. This research article is specifically built to generate Mathematica computer programming codes of exponentially fitted concurrent Milne's device (EFCMD) for solving special problems. Exponentially fitted concurrent Miln device is formulated via collocation/interpolation with power series as the approximate solution. Analyzing the EFCMD will produce the main local truncation error (MLTE) after showing the order, results were shown to demonstrate the functioning of Mathematica programming codes of EFCMD for resolving special problems at some selected bounds of convergence. The finished results were obtained with the assistance of Mathematica 9 kernel. Numerical results display that EFCMD do better than existing methods in terms of the maximum errors in the least studied bound of convergence as a result of varying/designing a suitable pace size, ascertain bound of convergence and error control.
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.
Mathematica Computer Codes of Exponential Concurrent Milne ’ S D Special Problems
2019
Over the years, scientific computing has contributed immensely to computational mathematics. Mathematica computer programming codes is known to provide computation and quick results. This research article is specifically built to generate Mathematica computer programming codes of exponentially fitted concurrent Milne’s device (EFCMD) for solving special problems. Exponentially fitted concurrent Miln device is formulated via collocation/interpolation with power series as the approximate solution. Analyzing the EFCMD will produce the main local truncation error (MLTE) after showing the order, results were shown to demonstrate the functioning of Mathematica programming codes of EFCMD for resolving special problems at some selected bounds of convergence. The finished results were obtained with the assistance of Mathematica 9 kernel. Numerical results display that EFCMD do better than existing methods in terms of the maximum errors in the least studied bound of convergence as a result of...
Study on Different Numerical Methods for Solving Differential Equations
Master of Science in Mathematics at University of Chittagong, 2011
This thesis paper is mainly analytic and comparative among various numerical methods for solving differential equations but Chapter-4 contains two proposed numerical methods based on (i) Predictor-Corrector formula for solving ordinary differential equation of first order and first degree (ii) Finite-difference approximation formula for solving partial differential equation of elliptic type. Two types of problems are discussed in detail in this thesis work, namely the ordinary differential equation in Chapters-2 & Chapter-3 and partial differential equations in Chapter-4. Also, Chapter-5 highlights the boundary value problems. The various chapters of this thesis paper are organized as follows: The chapters of this thesis paper are organized as follows: Chapter-1 of the thesis is an overview of differential equations and a brief discussion of their solutions by numerical methods. Chapter-2 deals with the solution of ordinary differential equations by Taylor’s series method, Picard’s method of successive approximation and Euler’s method. Derivation of Taylor’s series method with truncation error and application are discussed here. The solution of ordinary differential equations by Picard’s method of successive approximations and its application is discussed in detail. The definition of Euler’s method is mentioned, the simple pendulum problem is solved to demonstrate Euler’s method. Error estimations and geometrical representation of Euler’s method and the improved Euler’s method are mentioned as a Predictor-Corrector form, which forms being discussed in Chapter-3 next. Also in it, the comparison between Taylor’s series method and Picard’s method of successive approximation has given. Moreover the advantages and disadvantages of these three methods narrated in it. Chapter-3 provides a complete idea of the Predictor-Corrector method. Derivation of Milne’s predictor-corrector formula and Adams-Moulton Predictor-Corrector formula with their local truncation errors and applications are discussed here. Solutions of ordinary differential equations by the Runge-Kutta method with error estimation are studied in this chapter. Some improved extensions of the Runge-Kutta method are explained. Also, the general form of the Runge-Kutta method has given here. The law of the rate of nuclear decay is solved in this chapter by means of standard fourth-order Runge-Kutta method and then the obtained solution is compared with the exact solution, which is an application of the numerical method to the nuclear physics. Comparison between the Predictor-Corrector method and the Runge-Kutta method discussed in detail. Also, the advantages and disadvantages of these two methods discussed in it. Chapter-4 gives a review of the solution of partial differential equations. Three types of partial differential equations such as elliptic equations, parabolic equations and hyperbolic equations with methods of their solutions are discussed at length. To solve the method of the elliptic equation of iterations and relaxation are discussed. Schmidt's method and the Crank-Nicholson method are discussed to solve parabolic equations. The solution of vibrations of a stretched string is mentioned as a method of solution of hyperbolic equations. The solution of vibrations of the rectangular membrane by the Rayleigh-Ritz method has given here. A comparison between the iterative method and relaxation method has highlighted and then a total discussion of Rayleigh-Ritz with methods of iteration and relaxation reviewed in this chapter. Chapter-5 deals with the solution of the boundary value problems in both ordinary differential equations and partial differential equations. It provides a brief discussion of the finite-difference approximation method and shooting method with their applications. Also, the applications of Green’s function to solve boundary value problems are discussed in detail with the application. Moreover, the B-Spline method for solving two-point boundary value problems of order Four is introduced in this chapter at length. Derivations of cubic B-splines have represented. Cubic B-spline solutions of the special linear fourth-order boundary value problems, the general case of the boundary value problem, treatment of non-linear problems and singular problems have discussed here. Chapter-6 contains the proposal for the modification of two numerical methods. One of which proposed a modification of Milne’s predictor-corrector formula for solving ordinary differential equations of the first order and first degree, namely Milne’s (modified) Predictor-Corrector formula. One more step-length and one more term in Newton’s interpolation formula being calculated for deriving the predictor and corrector formulae of Milne’s (modified) Predictor-Corrector formula. Also, a modified formula for solving the elliptic equation by finite-difference approximation is proposed, namely surrounding 9-point formula. This formula is obtained by combining standard 5-point formula and diagonal 5-point formula, which gives a more contributive to find mesh points of a given domain in a certain region. Moreover, the advantages of proposed methods over previous methods are mentioned at the end of this chapter. Chapter-7 provides us the conclusions of this thesis paper. In this chapter, we have chosen the better methods in every chapter by comparing them. Also, the advantages and limitations of Milne’s (modified) predictor-corrector formulae and surrounding 9-point formula are given here. Finally, recommendations for future research and a list of few further works have mentioned.
A New Class of Highly Accurate Solvers for Ordinary Differential Equations
Journal of Scientific Computing, 2009
We introduce a new class of numerical schemes for the solution of the Cauchy problem for non-stiff ordinary differential equations (ODEs). Our algorithms are of the predictor-corrector type; they are obtained via the decomposition of the solutions of the ODEs into combinations of appropriately chosen exponentials, whereas the classical schemes are based on the approximation of solutions by polynomials. The resulting schemes have the advantage of significantly faster convergence, given fixed lengths of predictor and corrector vectors. The performance of the approach is illustrated via a number of numerical examples.
Formulating Mathematica pseudocodes of block-Milne's device for accomplishing third-order ODEs
International Journal of ADVANCED AND APPLIED SCIENCES
Formulating Mathematica pseudocodes for carrying out third-order ordinary differential equations (ODEs) is of essence necessary for proficient computation. This research paper is prepared to formulate Mathematica Pseudocodes block Milne's device (FMPBMD) for accomplishing third-order ODEs. The coming together of Mathematica pseudocodes and proficient computing using block Milne's device will bring about ease in ciphering, proficiency, acceleration and better accuracy. Side by side estimation and extrapolation is considered with successive function approximation gives rise to FMPBMD. This FMPBMD turns out to bring about the star local truncation error thereby finding the degree of the scheme. FMPBMD will be implemented on some numerical examples to corroborate the superiority over other block methods established by employing fixed step size and handled computation.
The BiM code for the numerical solution of ODEs
Journal of Computational and Applied Mathematics, 2004
In this paper we present the code BiM, based on blended implicit methods (J. the numerical solution of sti initial value problems for ODEs. We describe in detail most of the implementation strategies used in the construction of the code, and report numerical tests comparing the code BiM with some of the best codes currently available. The numerical tests show that the new code compares well with existing ones. Moreover, the methods implemented in the code are characterized by a diagonal nonlinear splitting, which makes its extension for parallel computers very straightforward.
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.
Solving Ordinary Differential Equations Using Parallel 2-Point Explicit Block Method
Mathematika, 2005
Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long series. A portable translator program accepts statements of the system of differential equations and produces a portable FORTRAN object code which is then run to solve the system. At each step of the integration, the object program generates the series for each component of the solution, analyzes that series to determine the optimal step, and extends the solution by analytic continuation. The translator is easy to use, yet it is powerful and flexible. The computer time required by this approach consists of time to run the translator plus time to run the object code, CPU time and storage requirements depend on the size and complexity of the system of ODEs. Theoretical estimates and empirical test results are given for Hull's test problems, and comparisons with DVERK and DGEAR from IMSL are given. The computer time for all preproeessmg, compilation, and linking Is included Taylor series method executes faster and yields a more accurate answer than the standard methods for most of the problems in the test set. The Taylor series method is most attractwe for small systems and for stringent accuracy tolerances.