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In this paper, we address a one-dimensional quasi-linear hyperbolic system of equations describing blood flow through compliant axi-symmetric vessels. From the symmetry analysis, we derive symmetry group of transformations and the corresponding symmetry generators by analyzing the parameters. Next, with the help of symmetry generators and invariant functions, we construct and classify the optimal system of subalgebras. Further, we obtained the similarity variables and similarity forms for each subalgebra leading to the reduction of the given governing coupled PDEs to the system of ODEs. Moreover, we studied the nature of blood flow velocity as well as the cross-sectional area of the arteries under the influence of arterial stiffness s graphically. Finally, the evolutionary behavior of weak discontinuity in the blood flow pattern is discussed with respect to aging.

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Using small amplitude wave theory, scattering of water waves by a vertical flexible barrier over a porous bed is studied. The boundary value problem in the form of Helmholtz equation is solved by the matched vertical eigenfunction expansion method. By exploiting the continuity of pressure and velocity at the interface along with Darcy's law for porous structure, the obtained coupled relation is solved by least-squares approximation method. The behavior of flexible barrier against the wave action for various physical quantities are studied and the numerical results are discussed. It is observed that due to the porous structure, a tranquility zone is created on the lee side of the barrier.