Research Projects

In computational Sciences, mathematical models are studied by finding the solutions of nonlinear equations. Examples include challenges in kinetic theory of gases, chemical reactor rate modeling, heat transfer, elasticity, and optimization, which often simplify to solution of nonlinear equations. Exact solutions of nonlinear equations are rarely found, thus iterative methods are very significant for approximating the solution of nonlinear equations. Some of the convergence properties that a reasonably good iterative method must possess are high convergence speed, less computational efficiency and wider domain of convergence etc. Thus, convergence of the sequence generated by iterative methods play an important role. Significant research has been conducted, leading to the development of iterative methods, along with their local, semilocal and global convergence analysis. Local convergence analysis require conditions on solution as well as on the operator and estimate the ball of convergence centered on the solution. Semilocal convergence analysis require conditions on initial guess as well as on the operator and estimate the ball of convergence centered on initial guess which allows us to guarantee the existence and uniqueness of the solution. Global convergence requires condition only on the operator and estimate the convergence domains which gives the convergence of iterative method starting from any point in this domain. To obtain domains of global convergence, we also assume conditions on auxiliary point as well as on the operator, this type of convergence is known as restricted global convergence. A limitation of local convergence is to impose conditions on the solution, which are often unknown or challenging to determine whereas in semilocal convergence, we must look for a good starting point that satisfies the assumed conditions. Nowadays, restricted global convergence and global convergence have become particularly significant since they impose no conditions on either the initial guess or the solution. Motivated by the recent developments in the field of iterative methods with and without derivatives, we present the convergence domains of iterative methods under various weaker continuity conditions. In order to show the applicability of our work, we will consider nonlinear Hammerstein integral equations as well as elliptic partial differential equation. The next aim of this proposal is to consider the nonlinear Hammerstein type integral equations with non-separable kernel. In this case, we can not apply the iterative method directly, so we first modify the iterative method and nonlinear integral equation. Very few authors have done the work in this direction. A key motivation for this proposed study is the potential to expand the convergence domain of higher-order iterative methods under weaker conditions.