In this paper, the optimal homotopy asymptotic method (OHAM) is applied to obtain an approximate solution of the nonlinear Riccati differential equation. The method is tested on several types of Riccati differential equations and comparisons that were made with numerical results showed the effectiveness and accuracy of this method.
In this paper, the homotopy-perturbation method (HPM) is applied to obtain approximate analytical solutions for the Cauchy reaction-diffusion problems. HPM yields solutions in convergent series forms with easily computable terms. The HPM is tested for several examples. Comparisons of the results obtained by the HPM with that obtained by the Adomian decomposition method (ADM), homotopy analysis method (HAM) and the exact solutions show the efficiency of HPM.
A map on a group is not necessarily an automorphism on the group. In this paper we study the necessary and sufficient conditions for a map on a non-split metacyclic p-group to be an automorphism, where we only consider p as an odd prime number. The metacyclic group can be defined by a presentation and it will be beneficial to have a direct relation between the parameters in the presentation and an automorphism of the group. We consider the action of an automorphism on the generators of the group mentioned. Since any element of a metacyclic group will be mapped to an element of the group by an automorphism, we can conveniently represent the automorphism in a matrix notation. We then use the relations and the regularity of the non-split metacyclic p-group to find conditions on each entry of the matrix in terms of the parameters in its presentation so that such a matrix does indeed represent an automorphism.
In this paper, we study the effects of symmetrization by the implicit midpoint rule (IMR) and the implicit trapezoidal rule
(ITR) on the numerical solution of ordinary differential equations. We extend the study of the well-known formula of Gragg
to a two-step symmetrizer and compare the efficiency of their use with the IMR and ITR. We present the experimental results
on nonlinear problem using variable stepsize setting and the results show greater efficiency of the two-step symmetrizers
over the one-step symmetrizers of IMR and ITR.
In this paper, we investigate the dynamical behavior in an M-dimensional nonlinear hyperchaotic model (M-NHM), where the occurrence of multistability can be observed. Four types of coexisting attractors including single limit cycle, cluster of limit cycles, single hyperchaotic attractor, and cluster of hyperchaotic attractors can be found, which are unusual behaviors in discrete chaotic systems. Furthermore, the coexistence of asymmetric and symmetric properties can be distinguished for a given set of parameters. In the endeavor of chaotification, this work introduces a simple controller on the M-NHM, which can add one more loop in each iteration, to overcome the chaos degradation in the multistability regions.
In this paper, systems of second-order boundary value problems (BVPs) are considered. The applicability of the homotopy-perturbation method (HPM) was extended to obtain exact solutions of the BVPs directly.
A map on a group is not necessarily an automorphism on the group. In this paper we determined the necessary and sufficient conditions of a map on a split metacyclic p-group to be an automorphism, where we only considered p as an odd prime number. The metacyclic group can be defined by a presentation and it will be beneficial to have a direct relation between the parameters in the presentation and an automorphism of the group. We considered the action of an automorphism on the generators of the group mentioned. Since any element of a metacyclic group will be mapped to an element of the group by an automorphism, we can conveniently represent the automorphism in a matrix notation. We then used the relations and the regularity of the split metacyclic p-group to find conditions on each entry of the matrix in terms of the parameters in its presentation so that such a matrix does indeed represent an automorphism.
This paper investigates and determines the solutions for the Diophantine equation x2 + 4.7b = y2r, where
x, y, b are all positive intergers and r > 1. By substituting the values of r and b respectively, generators of
x and yr can be determined and classified into different categories. Then, by using geometric progression
method, a general formula for each category can be obtained. The necessary conditions to obtain the
integral solutions of x and y are also investigated.
This paper presents an effective approach for the optimization of an in-feed centreless cylindrical grindingof EN52 austenitic grade steel (DIN: X45CrSi93) with multiple performance characteristics based on thegrey relational analysis. To study the effect of the entire space of the input variables, nine experimentalruns, based on the Taguchi method of L9 orthogonal arrays, were performed to determine the best factorlevel condition. The response table and response graph for each level of the machining parameters wereobtained from the grey relational grade. In this study, the in-feed centreless cylindrical grinding processparameters, such as dressing feed, grinding feed, dwell time and cycle time, were optimized by takinginto consideration the multiple-performance characteristics like surface roughness and out of cylindricity.By analyzing the grey relational grade, it was observed that dressing feed, grinding feed and cycle timehad significant effect on the responses. The optimal multiple performance characteristics were achievedwith dressing feed at level 1 (5 mm/min), grinding feed at level 2 (6 mm/min), dwell time at level 2(2.5 s), and cycle time at level 2 (11 s). It is clearly shown that the above performance characteristics inthe in-feed Centreless cylindrical grinding process can be improved effectively through this approach.
In this paper, we consider the system of Volterra-Fredholm integral equations
of the second kind (SVFI-2). We proposed fixed point method (FPM) to solve
SVFI-2 and improved fixed point method (IFPM) for solving the problem. In addition,
a few theorems and two new algorithms are introduced. They are supported by
numerical examples and simulations using Matlab. The results are reasonably good
when compared with the exact solutions.