Browsing by Author "Sattar, M. Abdus"

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Eulerian Method for Third Order Linear Systems
(University of Rajshahi, 2003) Haque, Md. Abdul; Sattar, M. Abdus
In this thesis, we have studied the solutions of the various types of third order linear systems of ordinary differential equations. In chapter I, we have considered the third order linear homogeneous system with constant coefficients and developed Eulerian methods for all possible cases of the characteristic roots of the variational matrix. In chapter 2, we have considered the third order linear nonhomogeneous system with constant coefficients. The method covers all the cases when the roots of the characteristic equation of the corresponding homogeneous linear system are real and distinct, real and equal and complex. In finding particular solutions for the nonhomogeneous system of equations we have used the method of variation of parameters. Finally, we have obtained solutions of this system with the help of Crammer's rule. In chapter 3, we have considered the generalized form of third order linear nonhomogeneous system with constant coefficients. We have extended the Eulerian method and developed new techniques for obtaining solutions of this system. In chapter 4, we have discussed the third order linear nonhomogeneous system with variable coefficients. This problem is very difficult to solve, so we have examined a special case of this problem. By using a suitabletransformation, we have reduced it to a third order linear nonhomogeneous system with constant coefficients and have found solutions by the method of Chapter 2. We have illustrated all the methods by several suitable examples.
KBM Asymptotic Method for third order Nonlinear Oscillations
(University of Rajshahi, 2000) Alam, Md. Shamsul; Sattar, M. Abdus
In most treatments or nonlinear oscillations hy perturbation method. only periodic oscillations are treated; transients are not considered. Krylov and Bogoliubov have used a perturbation method to discuss transients in the second order autonomous systems with small nonlinearities. The method is well known as an 'averaging method' in the theory of nonlinear oscillations. Later the method has been amplified and justified by L3ogoliubov and Mitropolskii. In this dissertation, we invt:stigatc SOllll' third order nonlinear oscillations based on the work of Krylov-13ogoliubov-Mitroplskii (KBJ\:1). First, nonlinear oscillation described by a third order ordinary autonomous differential equation is considered and a new perturbation tcclmiquc is developed. Then a method has been dcwlopcd 1(1 find asymptotic solution or a damped nonlinear system. The method is a generalization of Bogoliubov's asymptotic method and covers both under-damped and overdamped systems. Later clamped oscillations including critically damped motion have also been investigated in presence or more significant damping forces. Third order nonlinear oscillations with clamping and time delay, and with varying coenicients have been investigated separately. Moreover. a simple overdamped solution has been found for the third order weakly nonlinear systems.
On Some Approximate Solutions of Nonlinear Physical and Biological Problems
(University of Rajshahi, 2010) Uddin, Md. Alhaz; Alam, M. Shamsul; Sattar, M. Abdus
Most of the perturbation methods are developed to find the periodic solutions of nonlinear systems with small nonlinearities, transients are not considered. In 1947, first Russian scientists Krylov and Bogo Liubov introduced a perturbation method to discuss the transient' s response in the second order autonomous differential systems with small nonlinearities and this method is well known as "an asymptotic averaging method" in the theory of nonlinear oscillations. Later, this method has been amplified and justified by Bogo Liubov and Mitropolskii in 1961 and this extended method is known as the KBM method in literature. In this dissertation, we have presented an analytical technique based on He's homotopy perturbation technique and the extended form of the KBM method to investigate the solutions of second order strongly nonlinear physical and oscillating processes in biological systems with significant damping effects. Also, we have extended the KBM method to investigate the weakly third and fourth order nonlinear systems with slowly varying coefficients and damping effects. Firstly, second order damped nonlinear autonomous differential systems are considered and He's homotopy perturbation and the KBM methods have been extended to Duffing type strongly nonlinear physical problems with small damping effects. Then the method has been applied to find the analytical approximate solution of damped oscillatory nonlinear systems with slowly varying coefficients with strong nonlinearity. Further, this method has been developed to solve second order strongly nonlinear oscillating processes in biological system with small damping effects. We have also extended the homotopy perturbation technique to find the second approximation of second order strongly nonlinear differential systems with damping effects. We have extended the KBM method to determine the second approximation of third order weakly nonlinear damped oscillatory systems under some special conditions. Lastly, a unified KBM method has been presented to obtain the analytical approximate solution of a fourth order ordinary weakly nonlinear differential equation with varying coefficients and large damping, when a pair of eigen-values of the unperturbed equation is a multiple of the other pair or pairs. The methods have been illustrated by several examples.
On Some New Approximate Solutions of Fourth Order Nonlinear Differential Equations
(University of Rajshahi, 2005) Akbar, Md. Ali; Sattar, M. Abdus; Alam, M. Shamsul
Most of the perturbation methods are developed to find periodic solutions of nonlinear systems; transients are not considered. First Krylov and BogoLiubov introduced a perturbation method to discuss the transients in the second order autonomous systems with small nonlinearities. The method is well known as an "asymptotic averaging method" in the theory of nonlinear oscillations. Later, the method has been amplified and justified by BogoLiubov and Mitropolskii. In this dissertation, we have modified and extended the Krylov-. BogoLiubov-Mitropolskii (KBM) method to investigate some fourth order nonlinear systems. First a fourth order over-damped nonlinear autonomous differential system is considered and a new perturbation solution is developed. Then a method is developed to find asymptotic solution of damped oscillatory nonlinear systems. We then again solve the fourth order overdamped nonlinear systems under some special conditions. Later, unified KBM method is used to obtain the approximate solution of the fourth order ordinary differential equation with small nonlinearities, when a pair of eigen-values of the unperturbed equation is a multiple (I. e., double, triple etc.) of the other pair or pairs. In case of oscillatory processes some of the natural frequencies of the unperturbed equation may be in integral ratio and thus internal resonance is introduced, which is an interesting and important part of nonlinear vibrations. Modified and compact form of KBM method is used to find approximate solutions of fourth order nonlinear systems with large damping. The methods are illustrated by several examples.