Department of Mathematics
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Item A Study on Fixed Point Iterative Procedures(University of Rajshahi, 2016) Khatun, Miss. Saleha; Ali, M. ZulfikarFixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach’s fixed point theorem. Fixed point iterative procedures are one of the early achievements of fixed point theory for their usefulness to construct the solving technique of different nonlinear problems. Most of the physical problems of applied sciences and engineering are usually formulated as functional equations. Such equations can be written in the form of fixed point equations in an easy manner. It is always desired to develop an iterative procedure which approximates the solution of these equations in fewer numbers of steps. From this point of view, the main objective of our research is to fit a best fixed point iterative procedure whose working ability (rate of convergence) is better than that of the analogous fixed point iterative procedures. There exist a numeral number of fixed point iterative procedures in literature. But there raised a natural question that, “Which is the best fixed point iterative procedure under the equivalent situation?”. To find the answer of that question already many works have been completed by various renowned researchers; see for instance [12, 27, 37, 39, 47, 49] and their references. By the inspiration of these works here we have proposed a new three-step fixed point iterative procedure whose rate of convergence is better than that of analogous fixed point iterative procedures in case of contraction mapping. Using our new fixed point iterative procedure we have also established some weak and strong convergence theorems for non-expansive mapping and we apply these results to find the solutions of constrained minimization problems and feasibility problems. In the last part of our research, we have studied the fixed point iterative procedures with errors and proveda convergence theorem of multi-step Noor fixed point iterative procedure with errors for Zamfirescu operator, which generates the convergence theorems of rest fixed point iterative procedures with errors for the same operator.Item A Study on Interval Solutions of Nonlinear Systems(University of Rajshahi, 2011) Mollah, Md. Shirazul Hoque; Ali, M. ZulfikarSuppose we seek a solution of the nonlinear system f(x={f;(x1,x2, ..., Xn)} = 0 i = 1,2, ..., n where f1, f2, f3, ..., fn are continuous functions on an open set D in R”. There is good many methods for iterative interval solutions of system (I) for any such methods, R. E. Moore developed a technique for finding a safe starting point from which iterates converge, with a particular iterative method in mind, Krawczyk’s operator. Minoru Urabe established an existence and uniqueness theorem (1965) which helps verify the existence and uniqueness of an exact solution and to know the error bound to an approximate solution of a system like (1). His theorem assumes that all the computations are to be carried out in real numbers exactly. Our attempts will be made to combine M. Urabe's theorem and R. E Moore's technique theoretically as well as numerically considering interval version of Newton's method.Item An Analytical and Numerical Study on Fixed Point Theorems(University of Rajshahi, 2011) Asaduzzaman, Md.; Ali, M. ZulfikarFixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach's fixed point theorem. There exists a vast literature on the topic and this is a very active field of research at present. Let X be a set and let T be a mapping from X intoX. A fixed point of Tis an element x EX such thatT(x) = x. In mathematics, a fixed-point theorem is a result saying that a function T will have at least one fixed point, under some conditions on T that can be stated in general terms. In other words, a fixed-point theorem is a theorem that asserts that every function that satisfies some given property must have a fixed point. Fixed point theorems give the conditions under which maps (single or multivalued) have solutions. Fixed point theory is a beautiful mixture, of analysis, topology, and geometry. If we have an equation whose explicit solution is not so easy to find, in that case we rewrite the equation in the form T(x) = x to find its solution by applying any suitable fixed-point theorem. This method can be applied not only to numerical equations but also to equations involving vectors or functions. In particular, fixed-point theorems are often used to prove the existence of solutions to differential equations. Fixed point theorems also play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in social sciences, biology, chemistry, economics, engineering, physics and mathematics. For instance, the Banach [ 18, 40], Brouwer [ 18, 49] and Kakutani [ 18] fixed point theorems have been among the most-used tools in economics and game theory. Over the last 50 years the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena.Item An analytical method for finding exact Traveling wave solutions of some nonlinear Evolution equations (nlees) in biological and Mathematical problems(University of Rajshahi, Rajshahi, 2021) Ullah, Mohammad Safi; Ali, M. Zulfikar; Roshid, Harun-Or-Most of the natural happenings can be present by nonlinear modeling. The soliton theory is a highly effective section of nonlinear sciences that includes soliton, multi-soliton, rational, breather line, breather kinky, lump and rogue wave solutions. Such solutions are essential to realizing the internal properties of the nonlinear models. This dissertation presents exact traveling wave solutions of the three nonlinear models such as the (2+1) Bogoyavlenskii’s breaking soliton (BBS) equation, the (2+1)-dimensional Benjamin-Bona-Mahony-Burgers (BBMB) equation and the (3+1)-dimensional Sharma–Tasso–Olver-like (STOL) equation by applying Hirota bilinear method. By this method, we construct the bilinear form and find the interaction solutions of the above three models. We determine the multi-soliton and their interaction solutions of the BBS model and STOL model. Various properties of the obtained solutions are illustrated clearly with a number of 3D plot, 2D plot, density plot, curve plot and contour plot by choosing suitable parametric values via the computational software Maple 18.Item Analytical Methods to Investigate Exact Solutions for Space-Time Fractional Differential Equations Arising in the Real Physical Phenomena of Mathematical Physics and Biology(University of Rajshahi, Rajshahi, 2021) Rahman, Zillur; Ali, M. Zulfikar; Roshid, Harun-or-Fractional derivatives are most important to accurate nonlinear modeling of various real-world difficulties in applied nonlinear science and engineering incidents especially in the fields of crystal, optics and quantum mechanics even in biological phenomena. The investigation of exact solutions of such nonlinear models has great important to visualize the nonlinear dynamics. We consider the space-time fractional nonlinear differential equations for pulse narrowing transmission lines model, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM), complex Schrodinger and biological population models, the complex time fractional Schrodinger equation (FSE) and low-pass electrical transmission lines equation (ETLE) are studied with the effective unified method, Jacobi elliptic expansion function integral technique, generalized Kudryshov technique, modified simple equation (MSE) method respectively. As a result, we get some solitary wave solutions in the form of hyperbolic and combo hyperbolic-trigonometric :functions including both stable and unstable cases. We obtain kink wave, bright bell wave, dark bell wave, combo periodic-rogue waves, combo M-W shaped periodic-rogue waves in stable cases, and singular kink type in unstable solitonic natures. Lastly, we proposed an Improved Kudryashov method for solving any nonlinear fractional differential models. We apply the proposed approach to the nonlinear spacetime fractional model leading wave spread in electrical transmission lines (s-tfETL), the spacetime M-fractional Schrodinger-Hirota (s-tM-fSH) and the time fractional complex Schrodinger (tfcS) models to verify the effectiveness of the propose approach. The implementations of the introduced new technique on the models provide us periodic envelope, exponentially changeable soliton envelope, rational, combo periodic-soliton and combo rational-soliton solutions, which are much interesting phenomena in the nonlinear sciences. Beside the scientific derivation of the analytical findings, we represent the results graphically for clear visualization of the dynamical properties.Item Asymptotic Method for Time Dependent Nonlinear Differential Systems with Slowly Varying Coefficients(University of Rajshahi, 2013) Roshid, Harun-Or-; Ali, M. Zulfikar; Dey, PinakeeAlmost all perturbation methods are developed to find periodic solutions of nonlinear system where transients are not considered. First Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient process. In this dissertation, we have modified and extended the KBM method to investigate some fifth order and second order nonlinear systems in both cases with constant and slowly varying coefficients. At first, a fifth order damped nonlinear autonomous differential system is considered and a perturbation solution is developed. Then a procedure is developed for the same system with damped taking three of eigenvalues are real. After then we considered fifth order systems for over damped with small nonlinearity to obtain the transient response. We also developed a formula for fifth order critically damped nonlinear systems to control micro vibration, in micro and nano-technological industries that bring the system to equilibrium as quickly as possible without oscillating. After then we presented an analytical technique based on the extended Krylov-Bogoliubov-Mitropolskii method (by Popov) to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. At last, Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The implementations of the methods are illustrated by several examples.Item Damped Forced Vibration of Some Quasi-Linear Differential Systems(University of Rajshahi, 2008) Dey, Pinakee; Sattar, M A.; Ali, M. Zulfikar; Alam, M. ShamsulThere are many approaches for approximating solutions of nonlinear vibrating problems. The most common methods for constructing approximate analytical solutions to the nonlinear vibrating problems are the perturbation methods. These methods are developed to find only periodic vibrations of the nonlinear differential systems. In order to investigate the transients of nonlinear vibrations, 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 vibrations. Then the method was amplified and justified by BogoLiubov and Mitropolskii. These methods were applied to autonomous systems. Later, Arya and Bojadziev, Bojadziev and Hung, and Shamsul extended the Krylov-Bogo Liubov-Mitropolskii (KBM) method to sometime dependent nonlinear differential systems. In this dissertation, we extend the work of KBM and investigate some other time dependent non-linear differential systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear vibrations in presence of a slowly decaying external force. We then find an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the KBM method. Later, we find the perturbation solutions of damped forced vibrations using the modified KBM method, in which the coefficients change slowly varying with time. Further, this technique is used to obtain the second approximate solution of second order forced vibrations. Finally, this technique is used to obtain the higher approximate solution of an n-th order damped forced vibrating problem in the resonance case, and the stability of the stationary regime of vibrations has also been investigated. The methods are illustrated by several examples.Item Investigation of the Soliton and Multi-soliton Solutions of nonlinear evolution equations In Mathematical Physics(University of Rajshahi, Rajshahi, 2020) Hossen, Md. Belal; Ali, M. ZulfikarNonlinear evolution equations (NLEEs) play a noteworthy role in various scientific and engineering fields such as applied mathematics, plasma physics, fluid dynamics, optical fibers, biology, solid state physics, chemical physics, mechanics and geochemistry. Various effective procedure have been developed to solve NLEEs. In this work, we have discussed applications of two types methods: first type is modified double sub-equation (MDSE) method which is apply in the (1+1)-dimensional Burger equation, the (1+1)-dimensional Gardner equation and the (1+1)- dimensional Hirota-Ramani equation and secondly, Hirota’s Bilinear method which is apply in (2+1)-dimensional Breaking Soliton, the (2+1)-dimensional asymmetric Nizhnik-Novikov- Veselov equations, and (3+1)-D generalized B-type Kadomtsev-Petviashvili equation. Using Modified double sub-equation method, we have presented some complexiton solutions in terms of trigonometric, hyperbolic functions. Finally, the interaction phenomena of the achieved complexiton solutions between solitary waves and/or periodic waves are presented with in depth derivation. Based on the bilinear formalism and with the aid of symbolic computation, we determine multisolitons, breather solutions, rogue wave, lump soliton, lump-kink waves and multi lumps using various ansatze’s function. We notice that multi-lumps in the form of breathers visualize as a straight line. Besides this, the breather wave degenerate into a single lump wave is determined by using parametric limit scheme. Also, we reflect a new interaction solution among lump, kink and periodic waves via ‘rational-cosh-cos’ type test function. To realize dynamics, we commit diverse graphical analysis on the presented solutions. Obtained solutions are reliable in the mathematical physics and engineering.----