end{aligned} (3.2 The second order approximation of Caputo derivative as discussed in [13] is given as begin{aligned} left| r^{k+1}_{Delta t}right|le & EDelta t^{2}, end{aligned} (3.3 where E is a constant.
(ii) The 2-sided projection second order approximation H5 above, also used in [1, 8, 18, 22], is milder than the 1-sided projection second order approximation: (mathbf{H5^) : (Vert P_{k}(nabla_{xx}^{2}L_{bar{rho }}(x^{k},lambda^{k} -H_{k} bar{d}^{k} Vert =o( Vert bar{d}^{k} Vert )).
The 2-sided projection second order approximation H5 above, also used in [1, 8, 18, 22], is milder than the 1-sided projection second order approximation: (mathbf{H5^) : (Vert P_{k}(nabla_{xx}^{2}L_{bar{rho }}(x^{k},lambda^{k} -H_{k} bar{d}^{k} Vert =o( Vert bar{d}^{k} Vert )).
end{aligned} (14 The third order approximation to the exact solution is calculated using the improved adaptation proposed in the present paper.
(15) First-order approximation obigl varepsilon^{frac{1}{2}}bigr): quad L psi_{0})=0}
Figures 2 and 3 are plotted to show the admissible values of ħ for the function (f 0)) and (theta(0)) at 20th order approximation.
Table 10 Setting of parameters of the adaptive MEgPC approach for the biofilm model Degree of approximation p ˆ 5 Order of approximation p 3 Tolerance (time) TOL 5⋅10−6 Tolerance (elements) ε × Refinement threshold ρ 1 × Coarsening threshold ρ 2 ×.
Table 8 Setting of parameters of the adaptive MEgPC approach for the solution of RODE ( 1 ) Degree of approximation p ˆ (3,3) Order of approximation p 3 Tolerance (time) TOL 10−8 Tolerance (elements) ε 10−8 Refinement threshold ρ 1 1 Coarsening threshold ρ 2 4. In Figure 4 we plot the densities computed by SCM5, UKF and MEgPC at times t ∈ { 3, 000, 3, 500, 3, 550 }.
