A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

Beni-Suef University Journal of Basic and Applied Sciences

Table 1 The values of \(Hb_{i} (x)\) and its derivatives at the knots

	\({\boldsymbol{x}}_{i - 1}\)	\({\boldsymbol{x}}_{i}\)	\({\boldsymbol{x}}_{i + 1}\)
\(Hb_{i} (x)\)	\(\tau_{1}\)	\(\tau_{2}\)	\(\tau_{1}\)
\(Hb^{\prime}_{i} (x)\)	\(\tau_{3}\)	0	\(- \tau_{3}\)
\(Hb^{\prime\prime}_{i} (x)\)	\(\tau_{4}\)	\(- 2\tau_{4}\)	\(\tau_{4}\)

Where \(\tau_{1} = \sigma + \left( {1 - \sigma } \right)\frac{{\hat{\varsigma }_{2} - \hat{p}h}}{{2\left( {\hat{p}h\hat{\varsigma }_{1} - \hat{\varsigma }_{2} } \right)}}\), \(\tau_{2} = 1 + 3\sigma\), \(\tau_{3} = \frac{3\sigma }{h} + (1 - \sigma )\frac{{\hat{p}(\hat{\varsigma }_{1} - 1)}}{{2\left( {\hat{p}h\hat{\varsigma }_{1} - \hat{\varsigma }_{2} } \right)}}\), \(\tau_{4} = \frac{6\sigma }{{h^{2} }} + \left( {1 - \sigma } \right)\frac{{\hat{p}^{2} \hat{\varsigma }_{2} }}{{2\left( {\hat{p}h\hat{\varsigma }_{1} - \hat{\varsigma }_{2} } \right)}}\).