**6. Numerical Illustration**

The purpose of the present section is to demonstrate the convergence rate of the method. We will consider the maximum absolute error between the exact solution *u*(*xi*, *tk*) of the continuous problem and corresponding approximations *u<sup>k</sup> <sup>i</sup>* , which is given by

$$\varepsilon\_{\tau,h} = \max\_{0 \le i \le M, 0 \le k \le N} |u(\mathbf{x}\_i, t\_n) - u\_i^n|. \tag{89}$$

Moreover, we define the standard rates

$$
\rho^{\rm x}\_{\tau,h} = \log\_2 \left( \frac{\epsilon\_{\tau,2h}}{\epsilon\_{\tau,h}} \right), \& \quad \pounds \rho^{t}\_{\tau,h} = \log\_2 \left( \frac{\epsilon\_{2\tau,h}}{\epsilon\_{\tau,h}} \right). \tag{90}
$$

We consider the following multiterm time fractional delay sup-diffusion problem

$$\sum\_{r=0}^{2} p\_r \frac{\partial^{\mu\_r} u(\mathbf{x}, t)}{\partial t^{a\_r}} = \frac{(\mathbf{x} + 1)}{2} \frac{\partial^2 u}{\partial \mathbf{x}^2} + (\mathbf{x} + 1)^2 \frac{\partial u}{\partial \mathbf{x}} + f(u(\mathbf{x}, t), u(\mathbf{x}, t - 0.2), \mathbf{x}, t), \tag{91}$$

$$f(\mathbf{u}(\mathbf{x},t),\mathbf{u}(\mathbf{x},t-0.2),\mathbf{x},t) = -\mathbf{u}^2(t,\mathbf{x}) + \mathbf{u}(t-0.2,\mathbf{x}) + \mathbf{g}(\mathbf{x},t), \quad \forall (\mathbf{x},t) \in [0,1] \times [0,1]. \tag{92}$$

Note that *<sup>g</sup>*(*x*, *<sup>t</sup>*) is defined/derived, such that *<sup>u</sup>*(*x*, *<sup>t</sup>*) = *<sup>e</sup><sup>x</sup> <sup>x</sup>*2(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)2*<sup>t</sup>* <sup>3</sup> is the exact solution. The exact solution determines the initial condition and boundary conditions. The difference scheme (31) is employed in order to obtain the numerical solution. First, the numerical accuracy of this scheme in time will be verified. Taking a sufficiently small step size *h* and varying step size *τ*, the numerical errors and numerical convergence orders are listed in the lower half of Table 1. The computational results presented in Table 1 confirm the second-order convergence of the difference scheme (31) in time.

**Table 1.** Absolute errors and standard convergence rates in space and time when approximating the solution *u* of (1) with (*α*<sup>1</sup> = 1.3, *α*<sup>2</sup> = 1.5, *α*<sup>3</sup> = 1.7), while using the difference method (31). The parameters and conditions employed in this case correspond to those in Example 6.


Next, the numerical accuracy of the difference scheme in space for solving this example is examined. The numerical results of this scheme for different step sizes in space are calculated and the numerical errors, as well as the numerical convergence orders are recorded in the upper half of Table 1. Again, from which, one can find that, in this case the fourth-order convergence is achieved.
