## 4 Numerical Examples  
Here, we discuss some examples to explain the methodology of the proposed methods and conduct some comparison analysis. **Example 1**. : _Consider the two-dimensional linear fractional Fredholm integro-differential equation:_  
\[D_{t}^{0.5}u(x,t)=\frac{\sqrt{\pi}}{2}e^{x}-\frac{2(e-1)t}{3}+\int_{0}^{1} \int_{0}^{1}u(y,\tau)td\tau dy \tag{11}\]subject to the initial condition \(u(x,0)=0\). The exact solution is \(u(x,t)=\sqrt{t}e^{x}\). In this example, we find a numerical solution using the Haar wavelet method and then use the Laplace Haar wavelet method. In either approach, we consider the level of resolution \(J=4\)._  
_First_**, Approach:** _(Haar wavelet Method)_  
_We need to estimate the terms \(D_{t}^{0,5}u(x,t)\) and \(\int_{0}^{1}\int_{0}^{1}tu(y,\tau)d\tau dy\). Hence, we use_  
\[\frac{\partial}{\partial t}u(x,t)\approx\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{ i}(x)h_{j}(t). \tag{12}\]  
_Integrate both sides and use the initial condition \(u(x,0)=0\) to find that_  
\[u(x,t)\approx\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{i}(x)p_{j}(t). \tag{13}\]  
_Now, using the equally distance nodes \(x_{m}=\frac{m-0.5}{32}\) and \(t_{n}=\frac{n-0.5}{32}\), and using the estimates in (12) and (13), one can obtain the discrete system_  
\[\frac{1}{\sqrt{\pi}}\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{i}(x_{m})\int_{0}^{ t_{n}}h_{j}(\tau)(t_{n}-\tau)^{-0.5}d\tau=\frac{\sqrt{\pi}}{2}e^{x_{m}}-\frac{2(e-1 )t_{n}}{3}+\sum_{j=1}^{32}a_{ij}t_{n}p_{i}(1)q_{j}(1)\]  
_for \(n,m=1,2,3,...,32\), where \(q_{j}(x):=\int_{0}^{x}p_{j}(y)dy\). Use one of the common computer packages (we used Mathematica 11) to compute the Haar coefficients \(a_{ij}\), for \(n,m=1,2,3,...,32\). Figures 1-3 show the exact and approximate solutions for different levels of resolution._  
_Second Approach:_ _(Laplace Haar wavelet Method)_  
_We apply the Laplace transform to Equation (_11_) to get_  
\[\sqrt{s}U(x,s)=\frac{\sqrt{\pi}}{2}\frac{1}{s}e^{x}-\frac{2(e-1)}{3s^{2}}+ \frac{1}{s^{2}}\int_{0}^{1}\int_{0}^{1}u(y,\tau)d\tau dy\]  
_or, equivalently,_  
\[U(x,s)=\frac{\sqrt{\pi}}{2}\frac{1}{s^{3/2}}e^{x}-\frac{2(e-1)}{3s^{5/2}}+ \frac{1}{s^{5/2}}\int_{0}^{1}\int_{0}^{1}u(y,\tau)d\tau dy\]  
_Now, we take the inverse Laplace transform to obtain_  
\[u(x,t)=\sqrt{t}e^{x}-\frac{8(e-1)}{9}\sqrt{t^{3}}+\frac{4\sqrt{t^{3}}}{3 \sqrt{\pi}}\int_{0}^{1}\int_{0}^{1}u(y,\tau)d\tau dy.\]  
_To discretize the above equation, we take the nodes \(x_{n}=t_{n}=\frac{n-0.5}{32}\), \(n=1,2,3...,32\). By this choice, we have the system_  
\[\sum_{n=1}^{32}\sum_{m=1}^{32}a_{nm}h_{n}(x_{i})h_{m}(t_{j})=\sqrt{t_{j}}e^{x_ {i}}-\frac{8(e-1)}{9}\sqrt{t_{j}^{3}}+\frac{4\sqrt{t_{j}^{3}}}{3\sqrt{\pi}} \int_{0}^{1}\int_{0}^{1}\sum_{n=1}^{32}\sum_{m=1}^{32}a_{nm}h_{n}(y)h_{m}(\tau )d\tau dy.\]  
_Using the fact that \(\int_{0}^{1}h_{i}(x)dx=p_{i}(1)\) for all \(i=1,2,3,...,32\), one can write the above system as_  
\[\sum_{n=1}^{32}\sum_{m=1}^{32}a_{nm}h_{n}(x_{i})h_{m}(t_{j})=\sqrt{t_{j}}e^{x_ {i}}-\frac{8(e-1)}{9}\sqrt{t_{j}^{3}}+\frac{4\sqrt{t_{j}^{3}}}{3\sqrt{\pi}} \sum_{n=1}^{32}\sum_{m=1}^{32}a_{nm}p_{n}(1)p_{m}(1).\]  
_Figures 4 and 5 show the graphs of the approximate solution at the level of resolution \(J=4\)._Figure 1: The exact solution for Example 1. \[\frac{\partial}{\partial t}u(x,t)\approx\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{ i}(x)h_{j}(t). \tag{12}\]0 The approximate solution (blue) and the exact solution (red) for Example 1 at \(J=2\), \(t=0.046875\) (to the left) and at \(J=4\), \(t=0.046875\) (to the right) using HWM. \[\frac{\partial}{\partial t}u(x,t)\approx\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{ i}(x)h_{j}(t). \tag{12}\]1 The approximate solution for Example 1 at \(J=2\) (to the left) and at \(J=4\) (to the right) using HWM. **Example 2**. : _Consider the following problem_  
\[\frac{\partial}{\partial t}u(x,t)\approx\sum_{i=1}^{32}\sum_{j=1}^{32}a_{ij}h_{ i}(x)h_{j}(t). \tag{12}\]2  
_subject to the initial condition \(u(x,0)=0\), where \(\mathrm{sinc}\ (x)=\frac{\sin(\pi x)}{\pi x}\) is the normalized sinc function.