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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Mohammad Januar Ismail Burhan
Mathematics Study Program, Faculty of Science and Information Technology, Insitut Teknologi Kalimantan, Indonesia
Mathematics

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Spectral Analysis of a Cyclic Tank Mixing System Pebrudal Zanu; Sri Wahyu; Mohammad Januar Ismail Burhan
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.41466

Abstract

In this paper, we study a cyclic mixing tank system consisting of n identical tanks arranged in a closed-loop configuration, each operating under constant volume with equal inflow and outflow rates. The model is further generalized to an m-layer structure, where each layer comprises n circularly interconnected tanks. Based on mass balance principles, the concentration dynamics are formulated as a linear system of first-order ordinary differential equations. By exploiting the structured form of the system matrix, we derive an explicit analytical solution and obtain a closed-form characterization of its spectrum. We show that the eigenvalues of the system are explicitly given by k,j = (-2 + e2i(k-1)/n), which inherently guarantees the decay of all modes. In particular, the solution exhibits a polynomial-exponential form associated with the multilayer cyclic tank. We also establish that the system is exponentially stable under conditions ensuring decay of all modes. Finally, we present numerical simulations to validate the analytical results and to illustrate the transient dynamics of the multilayer system.
Co-Authors Pebrudal Zanu Sri Wahyu