IJAM: Volume 38, No. 1 (2025)

DOI: 10.12732/ijam.v38i1.3

DESIGNING OPTIMAL

SEQUENTIAL EXPERIMENTS FOR

CONVECTION-ADVECTION

MODELS USING FRACTIONAL

PARTIAL DIFFERENTIAL EQUATIONS

Edward L. Boone 1 , Ryad A. Ghanam 2

1 Department of Statistical Sciences and Operations Research

Virginia Commonwealth University

Richmond, VA 23284, USA

2 Department of Liberal Arts and Sciences

Virginia Commonwealth University in Qatar

Doha, QATAR

Abstract. Researchers are becoming increasingly interested in using Fractional Partial Differential Equation (FPDE) models for physical systems such

as modeling the flow of a gas through porous materials. These models rely on the fraction of the differentiation α, which needs to be estimated

from empirical data. Experimentation is required in order to generate empirical data, specifically, distance x from the pressure source and t the time

since the pressure was initially applied to the system which will generate an output pressure measurement p(x, t). While sampling times are easy to

choose when a sensor is in place, the location of sensors from the pressure source are typically arbitrarily chosen. This work shows how to design

experiments using a sequential design with a base design and sequentially adding sampling design points by finding the optimal sensor locations along x.

In this paper, we considered three methods of optimizations: A-optimality, D-optimality and E-optimality. For the A-optimality, we minimize the

of the sum of the marginal variances of all the parameters. For the D-optimality we maximize the determinant of the information matrix. For the

E-optimality, we o minimize the largest eigenvalue of the inverse of the information matrix. To estimate the parameters, a Bayesian framework

is utilized combined with a sequential design approach to search through the possible locations for the next sensor in the follow up design.

Two simple FPDE parameterizations are used to illustrate the method with an initial sensor location design of six sensors and with five additional

sensors locations determined sequentially. The simple examples suggest that the parameter values influence the location of the next best sensor location.

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How to cite this paper?
DOI: 10.12732/ijam.v38i1.3
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2025
Volume: 38
Issue: 1

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