# Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer

Yiannos Ioannou, Marios M. Fyrillas, Charalabos Doumanidis

Research output: Contribution to journalArticle

11 Citations (Scopus)

### Abstract

In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Péclet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate.

Original language English 1278-1283 6 Engineering Analysis with Boundary Elements 36 8 https://doi.org/10.1016/j.enganabound.2012.02.006 Published - Aug 2012

### Fingerprint

Heat and Mass Transfer
Asymptotic Solution
Fredholm Integral Equation
Linear regression
Integral equations
Approximate Solution
Mass transfer
Heat transfer
Regression analysis
Regression Analysis
Fluxes
Coefficient
Matched Asymptotic Expansions
Least Square Method
Variational Principle
Dimensionless
Length Scale
Strip
Integral Equations
Unknown

### Keywords

• Heat/mass transfer
• Integral equations
• Matched asymptotic expansions
• Regression analysis

### ASJC Scopus subject areas

• Analysis
• Applied Mathematics
• Computational Mathematics
• Engineering(all)

### Cite this

Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer. / Ioannou, Yiannos; Fyrillas, Marios M.; Doumanidis, Charalabos.

In: Engineering Analysis with Boundary Elements, Vol. 36, No. 8, 08.2012, p. 1278-1283.

Research output: Contribution to journalArticle

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AB - In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Péclet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate.

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