# Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation
## 2 Preliminaries of Fractional Calculus and Jacobi Polynomials
### Stability Analysis  
Firstly, we consider the linear case of the generalized fractional KdV equation as  
\[\left({}^{*}D_{a+}^{\alpha}u\right)\left(x,t\right)+\xi\frac{\partial u\left( x,t\right)}{\partial x}+\delta\frac{\partial^{3}u\left(x,t\right)}{\partial x^{3}}=f \left(x,t\right),\quad\left(x,t\right)\in\left(a,b\right)\times\left(0,T\right] \tag{32}\]  
and similarly, we discretize the linear case as  
\[\sum_{n=0}^{N}\left\{a_{k}\frac{1}{\eta_{k}}w_{k}J_{n}^{\gamma, \beta}\left(x_{j}\right)\left(b-x_{j}\right)+\xi\left[c_{1}J_{n-1}^{\gamma+1, \beta+1}\left(x_{j}\right)\left(b-x_{j}\right)-J_{n}^{\gamma,\beta}\left(x_{j }\right)\right]\right.\] \[\left.+\delta\left[c_{3}J_{n-3}^{\gamma+3,\beta+3}\left(x_{j} \right)\left(b-x_{j}\right)-c_{2}J_{n-2}^{\gamma+2,\beta+2}\left(x_{j}\right) \right]\right\}\phi_{n}\left(t_{k}\right)\] \[=\sum_{n=0}^{N}\frac{1}{\eta_{k+1}}\left[\sum_{i=1}^{k-1}\left(a_{ i+1}-a_{i}\right)w_{i}\phi_{n}\left(t_{i}\right)+a_{1}w_{0}\phi_{n}\left(t_{0} \right)\right]J_{n}^{\gamma,\beta}\left(x_{j}\right)\left(b-x_{j}\right)+f^{k} \left(x_{j}\right). \tag{33}\]  
Now we state our main theorem as follows. Theorem 1: _Assume that \(u\left(x,t\right)\) is a continuous function satisfying boundary conditions (11) for any \(t\), and \(f\left(x,t\right)\) is continuous function. Letting \(u_{N}^{k}\left(x\right)=\sum_{n=0}^{N}\phi_{n}\left(t_{k}\right)\psi_{n}\left(x\right)\) is the numerical solution of discrete scheme (33), the scheme (33) is stable unconditionally and for any \(k\geq 0\), it holds that_  
\[\|u_{N}^{k}\left(x\right)\|_{L^{2}}\leq\frac{w_{0}}{w_{k}}\|u_{N}^{0}\left(x \right)\|_{L^{2}}+\sum_{i=1}^{k-1}\frac{h_{i}}{w_{k}}\|f^{i}\|_{L^{2}}+\frac{ \eta_{k}}{a_{k}w_{k}}\|f^{k}\|_{L^{2}}, \tag{34}\]where \(h_{i}=\left(\frac{1}{a_{i}}-\frac{1}{a_{i+1}}\right)\eta_{i},\;\;i=1,\cdots,k-1\). Proof. : Let \(\hat{w}^{b,\gamma}\left(x\right)=\left(\frac{2}{b-a}\right)^{\beta+\gamma} \left(b-x\right)^{\beta}\left(x-a\right)^{\gamma}\), and \(v_{N-2}^{k}\left(x\right)=v_{N-2}\left(x,t_{k}\right)\) is a polynomial of degree \(N-2\) satisfying \(u_{N}^{k}\left(x\right)=v_{N-2}^{k}\left(x\right)\hat{w}^{b,\gamma}\left(x\right)\). The discrete form (30) at JGL nodes \(x_{j}\) can be rewritten as  
\[\frac{a_{k}}{\eta_{k}}w_{k}u_{N}^{k}\left(x_{j}\right)+\xi\partial_{x}u_{N+1}^ {k}\left(x_{j}\right)+\delta\partial_{xxx}u_{N}^{k}\left(x_{j}\right)=\sum_{i=1 }^{k-1}\frac{a_{i+1}-a_{i}}{\eta_{k+1}}w_{i}u_{N}^{i}\left(x_{j}\right)+\frac {a_{1}}{\eta_{k}}w_{0}u_{N}^{0}\left(x_{j}\right)+f^{k}\left(x_{j}\right). \tag{35}\]  
Multiplying both sides of (35) by \(v_{N-2}^{k}\left(x_{j}\right)\hat{w}_{j}\) and summing for \(j\) from \(0\) to \(N\) leads to  
\[\sum_{j=0}^{N}\left[\frac{a_{k}}{\eta_{k}}w_{k}u_{N}^{k}\left(x_{j }\right)+\xi\partial_{x}u_{N}^{k}\left(x_{j}\right)+\delta\partial_{xxx}u_{N}^ {k}\left(x_{j}\right)\right]v_{N-2}^{k}\left(x_{j}\right)\hat{w}_{j}\] \[=\sum_{j=0}^{N}\left[\sum_{i=1}^{k-1}\frac{a_{i+1}-a_{i}}{\eta_{k }}w_{i}u_{N}^{i}\left(x_{j}\right)+\frac{a_{1}}{\eta_{k}}w_{0}u_{N}^{0}\left( x_{j}\right)+f^{k}\left(x_{j}\right)\right]v_{N-2}^{k}\left(x_{j}\right)\hat{w}_{j},\]  
where \(\hat{w}_{j}\) is the corresponding weight.