Algorithms for Solving the Catch Equation Forward and Backward in Time

1982 ◽  
Vol 39 (1) ◽  
pp. 197-202 ◽  
Author(s):  
S. E. Sims
Keyword(s):  
Mortality Rate ◽  
Numerical Solution ◽  
Exponential Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Padé Approximation ◽  
Approximate Solutions ◽  
Fishing Mortality ◽  
Convergence Criteria ◽  
Pade Approximation

Approximate solutions to the catch equation for the fishing mortality rate both forward and backward in time are obtained with an application of the diagonal Padé approximation of degree four to the exponential function. In either case the resulting approximation as well as Pope's estimate are shown to serve quite well as starting values for Newton's Method which is used to obtain a numerical solution of the catch equation. Convergence criteria for Newton's Method are discussed in each setting.Key words: catch equation, Newton's method, Padé approximation, Pope's estimate

The Application of Newton’s Method to the Problem of Elastic Stability

1972 ◽  
Vol 39 (4) ◽  
pp. 1060-1065 ◽  
Author(s):  
E. Riks
Keyword(s):  
Numerical Solution ◽  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Elastic Stability ◽  
Auxiliary Equation

The numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

scholarly journals Approximate Solution of Riccati Differential Equation via Modified Greens Decomposition Method

2020 ◽  
Vol 70 (4) ◽  
pp. 419-424
Author(s):  
Amit Ujlayan ◽  
Mohit Arya
Keyword(s):  
Numerical Solution ◽  
Decomposition Method ◽  
Padé Approximation ◽  
Medical Science ◽  
Adomian Decomposition Method ◽  
Specific Class ◽  
Pade Approximation ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Adomian Decomposition

Riccati differential equations (RDEs) plays important role in the various fields of defence, physics, engineering, medical science, and mathematics. A new approach to find the numerical solution of a class of RDEs with quadratic nonlinearity is presented in this paper. In the process of solving the pre-mentioned class of RDEs, we used an ordered combination of Green’s function, Adomian’s polynomials, and Pade` approximation. This technique is named as green decomposition method with Pade` approximation (GDMP). Since, the most contemporary definition of Adomian polynomials has been used in GDMP. Therefore, a specific class of Adomian polynomials is used to advance GDMP to modified green decomposition method with Pade` approximation (MGDMP). Further, MGDMP is applied to solve some special RDEs, belonging to the considered class of RDEs, absolute error of the obtained solution is compared with Adomian decomposition method (ADM) and Laplace decomposition method with Pade` approximation (LADM-Pade`). As well, the impedance of the method emphasised with the comparative error tables of the exact solution and the associated solutions with respect to ADM, LADM-Pade`, and MGDMP. The observation from this comparative study exhibits that MGDMP provides an improved numerical solution in the given interval. In spite of this, generally, some of the particular RDEs (with variable coefficients) cannot be easily solved by some of the existing methods, such as LADM-Pade` or Homotopy perturbation methods. However, under some limitations, MGDMP can be successfully applied to solve such type of RDEs.

scholarly journals An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Grégory Antoni
Keyword(s):  
Approximate Solution ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Input Data ◽  
Numerical Technique ◽  
Nonlinear Function ◽  
Numerical Procedure ◽  
Approximate Solutions ◽  
First Order

This study concerns the development of a straightforward numerical technique associated with Classical Newton’s Method for providing a more accurate approximate solution of scalar nonlinear equations. The proposed procedure is based on some practical geometric rules and requires the knowledge of the local slope of the curve representing the considered nonlinear function. Therefore, this new technique uses, only as input data, the first-order derivative of the nonlinear equation in question. The relevance of this numerical procedure is tested, evaluated, and discussed through some examples.

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