Tridiagonal matrix algorithm

Abstract: An approach is presented to solve bordered tridiagonal linear equations (BTLES) By the approach, a BTLE is first converted into three or more tridiagonal linear equations (TLES) that are independent each other, then the solution of the BTLE can be obtained via the solutions of the TLES Since the TLES are independent each other, their solution can be obtained via parallel computing under heterogeneous environments The approach costs at most O(n2) of time complexity in sequential mode Detail mathematical deduction is presented to reveal the approach and a framework is introduced to implement the approach The approach, which can also be available for grid computing, greatly increases the flexibility and agility of computations as well as the computational efficiency

Abstract: In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor's series, provided the shift is of $o(\varepsilon )$. Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then Trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, $\delta $ and perturbation parameter, $\varepsilon $. Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed.