  Wheeler (1964) had formulated Mach's principle as the boundary condition for
general relativistic field equations. Here, we use this idea and develop a
modified dynamical model of cosmology based on imposing Neumann boundary
condition on cosmological perturbation equations. Then, it is shown that a new
term appears in the equation of motion, which leads to a modified Poisson
equation. In addition, a modified Hubble parameter is derived due to the
presence of the new term. Moreover, it is proved that, without a cosmological
constant, such a model has a late time-accelerated expansion with an equation
of state converging to $w < -1$. Also, the luminosity distance in the present
model is shown to differ from that of the $\Lambda CDM$ model at high
redshifts. Furthermore, it is found that the adiabatic sound speed squared is
positive in radiation-dominated era and then converges to zero at later times.
Theoretical implications of the Neumann boundary condition have been discussed,
and it is shown that, by fixing the value of the conjugate momentum (under
certain conditions), one could derive a similar version of modified dynamics.
In a future work, we will confine the free parameters of the Neumann model
based on hype Ia Supernovae, Hubble parameter data, and the age of the oldest
stars.
