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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.

In general, through a class of numerical methods known under the aliases boundary integral, boundary element, boundary integral-equation, panel and Green's function methods, the classical scalar transport process in a medium [1] can be described by a kernel function (Green's function G), which depends on a number of parameters (Pi) characterizing the material/process conditions, and the flux distribution (q) [2] and [3]. The resulting equation is a convolution integral equation known as a Fredholm integral equation [4]. This class of numerical methods offers the natural choice for inverse transport problems as it combines numerical simplicity and accuracy. Regarding the former, the reduction of the differential equation to an integral equation over the boundary reduces the dimensionality, hence the complexity, of the problem. Furthermore, the integral representation allows the consideration of an infinite-domain, direct calculation of the concentration gradient, and de-singularization of the singular points which translate into a high degree of accuracy [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16] and [21]. Although the resulting integral equation is linear, it is notoriously difficult to solve and explicit solutions are only available for some special forms of the kernel [31]. Techniques for obtaining asymptotic solutions have appeared in the literature [8], [9], [10], [11], [12], [42], [43], [14], [17], [18], [19] and [20] however, it is often difficult to proceed to higher order or to obtain a matched asymptotic expansion. In natural and technological processes described by scalar (mass, energy, charge, etc.) transport phenomena, inverse transport problems have recently re-gained the keen interest of the scientific and engineering communities. Such inverse transport formulations seek the requisite input flux distribution (q), as well as the total flux (Q) required, in order to obtain a specified density distribution in the continuum (e.g. concentration, temperature, potential, etc.). The reason for this rekindled interest is due to both technical and mathematical progress in transport research: Technological means have become available over the past few decades for control of the flux distribution q, via high-bandwidth continuous scanning sources of lower dimensionality (e.g. robotic or servomechanism-guided plasma or laser heat torches [22], [23] and [24], high-density infrared lamp strips [25], scanned electron or ion implantation beams [26], etc.). In addition, highly localized discrete stationary sources, at dimensions down to the nanoscale (e.g. nanoheaters [27] and [28]) and in custom-designed distributions or addressable multiplexed array configurations, are presently introduced for precise implementation of transport actuation in a variety of applications (self-heating, self-repaired materials, etc.). From the computational perspective, aside from off-line numerical simulation formulations (finite difference, finite element, boundary value methods, etc.) and useful software tools that have become ubiquitous in engineering practice, there is also renewed attention to analytical techniques for inverse transport problems. Analytical solutions afford unique physical insights on the explicit effects of transport parameters, as well as computational efficiencies enabling their in-process technical implementation, in conjunction with the new actuation technologies mentioned above. Typical such analytical methods involve an optimization approach in approximating the desired density distribution in the continua by the action of combined/distributed elementary influxes (e.g. via a Green–Galerkin technique [29]). However, in an effort to avoid real-time numerical computation costs as much as possible during transport applications, there is a need for efficient optimization in the solution of the Fredholm integral equations (describing linear, stationary transport in a low-dimensionality continuum) by utilization of known analytical distribution functions in the asymptotic limits of the process parameters, i.e. corresponding to simpler, ideal physical problems. Motivated by such a necessity, in the next Section 2 we develop a method that can improve the existing asymptotic solutions of Fredholm integral equations of the first kind, through a least squares regression method. In the following Section 3, the method is implemented in a number of examples motivated from classical problems in heat and mass transfer.

We have developed a method that improves and extends existing asymptotic solutions of Fredholm integral equations of the first kind. The method relies on describing the unknown function, i.e. the flux distribution, through a linear combination of known asymptotic approximations of the integral equation. Subsequently, the integral equation is reformulated as a variational principle, i.e. least squares regression, and explicit expressions for the flux distribution can be obtained. The method has been applied to fundamental heat and mass transfer problems presented in the literature. The results suggest that the method is quite accurate, efficient, versatile and scalable. It has been successfully applied to problems with multiple parameters and to problems where limited asymptotic solutions are available. It is tempting to infer that the improved asymptotic solution is a matched asymptotic expansion, and that matching through least squares regression may not be limited to integral equations. More importantly, given that the integral equation formulation is the natural choice for formulating the inverse transport problem, and that the method provides tunable precision (via employing higher-order terms as needed), these make it invaluable for in-process actuation control at a trivial computational cost, especially for low-bandwidth (slow) transport processes. Towards such real-time implementation of the method in technical applications, two related problems will be attacked in future work. First, inevitable process disturbances and parameter alterations in real engineering settings are reflected in the current formulation as uncertainties in the kernel Green's function, i.e. in the parameters of the asymptotic expressions eventually affecting the parameters in the least square solution. An approach dual to the control problem of actuation input addressed above, i.e. the observation of resulting output distributions for in-process identification of the requisite uncertain process parameters, will be tacked using similar least-squares optimization algorithms. Such a real-time observer/identification technology is enabled by technological progress in non-intrusive, non-destructive sensory measurement methods, e.g. infrared pyrometry, ultrasound, x-ray scanning microtomography, etc. Second, for relatively higher-bandwidth (faster) transport conditions, a transient formulation of the method will be developed by addressing Fredholm integral equations of the second kind through appropriate least-squares optimization dynamic formulation and dynamic asymptotic expressions. These two future developments will enable application of the inverse/control methodology to manufacturing processes, such as selective laser sintering, etc.