Invariant Painleve analysis and coherent structures of two families of reaction-diffusion equations

Exact closed-form coherent structures (pulses/fronts/domain walls) having the form of complicated traveling waves are constructed for two families of reaction-diffusion equations by the use of invariant Painleveacute analysis. These analytical solutions, which are derived directly from the underlying PDE\u27s, are investigated in the light of restrictions imposed by the ODE that any traveling wave reduction of the corresponding PDE must satisfy. In particular, it is shown that the coherent structures (a) asymptotically satisfy the ODE governing traveling wave reductions, and (b) are accessible to the PDE from compact support initial conditions. The solutions are compared with each other, and with previously known solutions of the equations

An approximate analytic solution of a particular boundary value problem

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the whole x′y′-plane and finite in z′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution

Coherent Structures Of The Φ\u3csup\u3e4\u3c/sup\u3e Equation Via Invariant Painlevé Analysis

Exact closed-form coherent structures (pulses/fronts/domain walls) having the form of complicated traveling waves are constructed via invariant Painlevé analysis for the Φ4 equation, which belongs to the family of Klein-Gordon equations. These analytical solutions, which are derived directly from the underlying PDE\u27s. are investigated in the light of restrictions imposed by the ODE that any traveling wave reduction of corresponding PDE must satisfy. In particular, it is shown that the coherent structures a) asymptoticaly satisfy the ODE governing traveling wave reductions, and b) are accessible to the PDE from compact support initial conditions. The solutions are compared with each other, and with previously known solutions of the equation

Analytical Prediction Of Homoclinic Bifurcations Following A Supercritical Hopf Bifurcation

An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the pe riodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely

In Vitro Evaluation of Microleakage and Microhardness of Ethanolic Extracts of Propolis in Different Proportions Added to Glass Ionomer Cement

Objective: To evaluate the effect of ethanolic extracts of propolis (EEP) addition in different proportions to glass ionomer cement (GIC) on microleakage and microhardness of GIC. Study design: The cement was divided into four groups: one using the original composition and three with 10%, 25%, and 50% EEP added to the liquid and then manipulated. For microleakage assessment, sixty primary molars were randomly divided into four groups (n=15). Standard Class II cavities were prepared and then filled with EEP in different proportions added to GICs. Microleakage test was performed using a dye penetration method The data were analyzed using one-way ANOVA and Mann - Whitney U tests (alpha = 0.05). Disc shaped specimens were prepared from the tested GIC to determine Vickers hardness (VHN). The data were analyzed using one-way ANOVA and post hoc Tukey test (alpha = 0.05). Results: There were no statistically significant differences between the groups in terms of microleakage (p > 0.05). There were statistically signcant differences between the VHN values of groups (p < 0.05). Increasing addition of EEP to GIC statistically significantly increased VHN value of GIC (p < 0.05). Conclusions: The addition of EEP to GIC increased the microhardness of the GIC and did not adversely affect the microleakage. Thus, it might be used during routine dental practice due to its antibacterial propertie