We can apply a step-by-step method to the initial value problem of μ equations d u ( t ) d t = A ( u ( t ) ) u ( t ) + b ( u ( t ) ) + s ( t ), with the initial condition (2), u(x i, y j, 0) = U0(x i, y j ), and computes the approximation un+1to using1) using the approximate solution u n at the time level t n, with t n = n Δt and Δt the time step.
In several literature works (Odibat et al. in Comput. Math. Appl. 59 1462-1472, 2010; Alomari in Comput. Math. Appl. 61 2528-2534, 2011; Gökdoğan et al. in Math. Comput. Model. 54 2132-2138, 2011), authors have updated an initial condition in each sub-domain by using the approximate solution in the previous sub-domain.
The alternative approach is to use the approximate solution itself in order to find such an estimate.
Thus, it is not appropriate to use the approximate solution for interpreting the results of an infiltration tracer test.
In recent years, the numerical studies for evaluating the behavior of machine vibration foundation have been conducted, but the design using the approximate analytical solution [9] still has been made a lot in the practice.
Instead of the exact phenomenological equations the Nernst-Planck equations can be used for the approximate solution.
The finite-difference method with square grids is used for the approximate solution in the 'nonsingular' part, and in the blocks the integral representations of the harmonic function are approximated by the exponentially convergent mid-point quadrature rule (see [9]).
Then the LP formulation as in (4) is used to calculate the approximate solutions.
The theorem is then used to study the approximate solutions of the functional equation Ax = y.
The fast Fourier transform and the inverse fast Fourier transform are used to compute the approximate solutions, and all computations are done in Matlab 7.0.
