We present solutions to both trifurcated and pentafurcated spaced waveguides using the mode matching (or eigenfunction expansion) method. While the trifurcated problem with mean fluid flow has been solved previously using the Wiener-Hopf technique, we solve this problem to validate and demonstrate our method. We then show how we can easily generalize the method to the pentafurcated problem that has not been solved previously. We observe that mode matching method is easier to derive and generalize than the Wiener-Hopf technique. We also investigate the numerical solution in detail for various geometries to model practical exhaust systems.

4This paper describes the diffraction of the lowest plane wave prop-5 agating out of the opening of a semi-infinite hard duct which is sym-6 metrically placed inside an infinite soft duct. The whole system forms 7 a pentafurcated waveguide whose solution is given by eigenfunction ex-8 pansion method. Same method has been used to validate the results 9 of a trifurcated duct problem previously tackled by using the standard 10 Wiener-Hopf technique. Some graphical results showing the influence of 11 waveguide spacing on the reflection coefficient are presented. We com-12 pared our results with the related existing work. We observed that the 13 accumulative value of reflection for current pentafurcated duct is 1375.1 14 which is greater among the related trifurcated and existing pentafurcated 15 ducts. Hence the soft lining on outer plates gives better results to atten-16 uate the unwanted noise. These problems have application in acoustics.

The phenomenon of extraordinary acoustic transmission (eat) in a resonator, which has recently been investigated experimentally, is studied theoretically. It is shown that the combination of a single propagating mode and a symmetry orthogonal to the direction of propagation for a resonator leads to eat. This is accomplished by decomposing the problem using symmetry, the Blaschke product and the properties of functions of a single complex variable which have modulus one on the real axis. The conditions of a resonator requires that the solution has singularities in the analytic extension to complex frequencies (resonances) and it is precisely near these resonances that we observe eat. The condition of a Blaschke product requires that there is a zero at the complex conjugate of the singularity and eat occurs when the solution on the real axis passes between these complex conjugate pairs of poles and zeros. A detailed numerical study of the problem is conducted and we show that once the single mode of propagation or the symmetry is broken then eat (at least perfect transmission) no longer holds generally.

A mathematical model which is non-linear in nature with non-integer order φ, 0 < φ ≤ 1 is presented for exploring the SIRV model with the rate of vaccination μ 1 and rate of treatment μ 2 to describe a measles model. Both the disease free F 0 and the endemic F * points have been calculated. The stability has also been argued for using the theorem of stability of non-integer order differential equations. R 0 , the basic reproduction number exhibits an imperative role in the stability of the model. The disease free equilibrium point F 0 is an attractor when R 0 < 1. For R 0 > 1, F 0 is unstable, the endemic equilibrium F * subsists and it is an attractor. Numerical simulations of considerable model are also supported to study the behavior of the system.