On the differential transform method of solving boundary eigenvalue problems: An illustration

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 101; no. 5
• Main Authors: Narayana, M., Shekar, M., Siddheshwar, P.G., Anuraj, N.V.
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01-05-2021
• Subjects: Boundary conditions; boundary eigenvalue problem; Convergence; Darcy–Brinkman convection; Differential equations; differential transform method; Eigenvalues; general boundary conditions; Ordinary differential equations; porous medium; Power series; Taylor series
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.202000114

The differential transform method (DTM) is a simple technique based on the Taylor series. Applying the DTM, a given linear boundary eigenvalue problem (BEVP) involving ordinary differential equations is converted into a recurrence relation or a system of recurrence relations for the Taylor coefficients. This ultimately leads to the solution of the problem in the form of an infinite power series with an appropriate region of convergence. The present paper aims to apply the DTM in solving a BEVP arising from the Darcy–Brinkman convection in a rectangular box subject to general boundary conditions whose vertical sidewalls are assumed to be impermeable and adiabatic. The non‐dimensional temperature difference between the plates represented by the Darcy–Rayleigh number, the eigenvalue of the problem, is obtained as a function of the width of the Bénard cell (2πb : b is the horizontal wave number) and other parameters using the DTM. The work includes investigation on the convergence of the series solution. The solution by the DTM is compared with that obtained by the MATLAB bvp4c routine and excellent agreement is found thereby establishing the accuracy of the DTM.
The differential transform method (DTM) is a simple technique based on the Taylor series. Applying the DTM, a given linear boundary eigenvalue problem (BEVP) involving ordinary differential equations is converted into a recurrence relation or a system of recurrence relations for the Taylor coefficients. This ultimately leads to the solution of the problem in the form of an infinite power series with an appropriate region of convergence….