Journal of Pure and Applied Mathematics: Advances and ApplicationsAuthor's: Imanova Mehrivan and Vagif Ibrahimov
Pages: [25] - [48]
Received Date: May 7, 2024
Submitted by:
DOI: http://dx.doi.org/10.18642/jpamaa_7100122301
Specialists have always dealt with approximate calculations. However, official recognition was right in Newton’s era. For example, Newton-Cottes method for calculation of definite integrals. Followers of Newton, have constructed approximate methods for solving some mathematical problems. Leonard Euler has constructed his famous direct method for solving ODEs (Ordinary Differential Equations), which is refined and developed in the works of Adams-Moulton, Adams-Bashford, Cowell and others. All these methods are generalize in the results of which has constructed multistep methods with the constant coefficients. We have described here the chronological development of numerical methods. Noted that multistep methods with constant coefficients have investigated by many authors as Shura-Bura, Bakhalov, Dahlquist, Lambert, Ibrahimov, Henricy, Brunner, Imanova and etc. This method has generalized by some authors: Dahlquist, Enrite, Kobza, Ibrahimov and etc. In the works of named authors have shown that, if in the multistep methods participate the derivatives of the desired functions, then these methods will be more exact. By using this idea, some authors proposed to reduce the differential equation of the first order to a differential equation of any order and then use the appropriate method using high derivatives of desired solutions. By taking into account these many authors decided to use multistep multiderivative methods. Noted that these methods can be used in different form depending from the order of differential equations. In the construction of multistep methods in usually is used the presentation of the right-hand side of the given differential equation. Consequently, here for each of which it turns out numerous special cases, needed to construct and explore multistep multiderivative methods. Therefore, here consider to investigation of Multistep Multiderivative Methods (MMM) explicit, implicit and advanced types and their application for solving of initial value problem for ODEs and for the Volterra integro-differential equation and also application of the above mentioned methods for solving of the Volterra integral equation of the second kind.
multistep multiderivative methods, initial-value problem for both ODEs and the Volterra integro-differential equations, the Volterra inteqral equations, advanced and hybrid methods, stable and l-stable, degree of optimal methods.
