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Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.

Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some algorithms where four discrete forms of the Caputo derivative and three different numerical techniques of solving ordinary differential equations are proposed. We then illustrate how to introduce classical initial conditions into equations where the Riemann-Liouville derivative is included.

The purpose of these lecture notes is to provide an introduction to computational methods for the approximate solution of ordinary di_erential equations (ODEs).Only minimal prerequisites in di_erential and integral calculus, di_erential equation theory, complex analysis and linear algebra are assumed. The notes focus on the construction of numerical algorithms for ODEs and the mathematical analysis of their behaviour, covering the material taught in the M.Sc. in Mathematical Modelling and Scienti�c Computation in the eight-lecture course Numerical Solution of Ordinary Di_erential Equations.

This report describes the derivation of a set of ordinary differential equations to model radical chain polymerization. These equations have the mathematical property of stiffness and are difficult to solve numerically. We show how these equations can be solved efficiently using either the Gear or Kaps Rentrop method. We also show how the kinetic scheme can be expanded to allow for the presence of contaminant scavenger molecules, and we apply these schemes to model experimental results for the polymerization of N- vinyl-2-pyrrolidone obtained from dilatometry measurements.

A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well known and popular concept in mathematics because of its massive application in real world problems. Differential equations are one of the most important mathematical tools used in modeling problems in Physics, Biology, Economics, Chemistry, Engineering and medical Sciences. Differential equation can describe many situations viz exponential growth and de cay, the population growth of species, the change in investment return over time. We can solve differential equations using classical as well as numerical methods, In this paper we compare numerical methods of solving initial valued first order ordinary differential equations namely Euler method, Improved Euler method, Runge Kutta method and their accuracy level. We use here Scilab Software to obtain direct solution for these methods. Vibahvari Tukaram Dhokrat "Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd45066.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/45066/comparative-analysis-of-different-numerical-methods-for-the-solution-of-initial-value-problems-in-first-order-ordinary-differential-equations/vibahvari-tukaram-dhokrat

A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well known and popular concept in mathematics because of its massive application in real world problems. Differential equations are one of the most important mathematical tools used in modeling problems in Physics, Biology, Economics, Chemistry, Engineering and medical Sciences. Differential equation can describe many situations viz exponential growth and de cay, the population growth of species, the change in investment return over time. We can solve differential equations using classical as well as numerical methods, In this paper we compare numerical methods of solving initial valued first order ordinary differential equations namely Euler method, Improved Euler method, Runge Kutta method and their accuracy level. We use here Scilab Software to obtain direct solution for these methods. Vibahvari Tukaram Dhokrat "Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd45066.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/45066/comparative-analysis-of-different-numerical-methods-for-the-solution-of-initial-value-problems-in-first-order-ordinary-differential-equations/vibahvari-tukaram-dhokrat

Series of computer subroutines integrates systems of ordinary differential equations and is used for numerical quadrature.

Study compares the use of five different methods for the computer solution of the restricted three-body problem. It describes the implementation of each method on a burroughs B-5000 computer and in terms of speed and accuracy.

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A method by which a particular set of coupled ordinary nonlinear differential equations may be solved numerically is presented. The nonlinearity is present due to the quadratic terms present in the derivatives. An example is given to illustrate the problem when applied to a model of a cable towed body moving through a viscous fluid.

This is a survey article which deals with the advantages of using interval methods for the numerical solution of initial value problems for ordinary differential equations. (Author)

In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of differential equation of integer order connected with inverse forms of Abel-integral equations. The algorithm is used for solution of the linear and non-linear equations.

The theoretical analysis and computational implementation of factorization-based methods for the numerical solution of linear boundary value problems for ordinary differential equations are presented. The methods are optimal with respect to certain clearly defined criteria. Numerical examples show the effectiveness of a general code based on the factorization method. Direct methods are characterized by the solution of global (linear) algebraic systems for the discrete solution. In this sense multi-shooting may be regarded as a hybrid method between the two classes. A sophisticated recent example of multi-shooting is the BOUNDPAC package. Indirect solution methods are characterized by the association of the boundary value problem with certain auxiliary initial value problems (IVP). The auxiliary IVP's are generally solved uni-directionally (forward), but a subclass of initial value based methods are based on bi-directional (double sweep) strategies.

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

Bibliography: p. 79-80

Bibliography: p. 79-80

Bibliography: p. 79-80

Bibliography: p. 79-80