Numerical solution of high order linear complex differential equations via complex operational matrix method. (English) Zbl 1441.65058
The authors consider a higher order linear nonhomogeneous differential equation having analytic coefficients in a rectangle in the complex plane
\[
\sum_{k=0}^m\omega_k(z)y^{(k)}(z)=g(z),
\]
with initial conditions
\[
\sum_{n=0}^{m-1}e_{nk}y^{(k)}(0)=\lambda_{k},\quad k=0, 1, \dots, m-1,
\]
where \(e_{nk}\) and \(\lambda_{k}\) are constants. By means of the method of orthonormal Bernstein polynomials, they obtain a complex operational matrix of differentiation.
The advantage of the proposed method is that complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative methods. Further, the authors discuss convergence analysis of the proposed method and they establish an upper error bound under weak assumptions. Some numerical examples of second order cases are given.
The advantage of the proposed method is that complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative methods. Further, the authors discuss convergence analysis of the proposed method and they establish an upper error bound under weak assumptions. Some numerical examples of second order cases are given.
Reviewer: Katsuya Ishizaki (Chiba)
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 34M03 | Linear ordinary differential equations and systems in the complex domain |
Keywords:
complex differential equations; orthonormal Bernstein polynomials; operational matrix method; error analysisReferences:
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