This study shows how to obtain least-squares solutions to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by any existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a constrained expression, derived from Theory of Connections. In this expression, the differential equation constraints are embedded and are always satisfied. The resulting constrained expression is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10 −12. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.

This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second order differential equations. The proposed method has two steps. The first step consists of writing a constrained expression, introduced in Ref. [1], that has embedded the differential equation constraints. These expressions are given in term of a new unknown function, g(t), and they satisfy the constraints, no matter what g(t) is. The second step consists of expressing g(t) as a linear combination of m independent known basis functions, g(t) = ξ T h(t). Specifically, Chebyshev orthogonal polynomials of the first kind are adopted for the basis functions. This choice requires rewriting the differential equation and the constraints in term of a new independent variable, x ∈ [−1, +1]. The procedure leads to a set of linear equations in terms of the unknown coefficients vector, ξ, that is then computed by least-squares. Numerical examples are provided to quantify the solutions accuracy for initial and boundary values problems as well as for a control-type problem, where the state is defined in one point and the costate in another point.

Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the Theory of Functional Connections (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches. IntroductionDifferential equations (DE) and their solutions are important topics in science and engineering. The solutions drive the design of predictive systems models and optimization tools. Currently, these equations are solved by a variety of existing approaches with the most popular based on the Runge-Kutta family [1]. Other methods include those which leverage low-order Taylor expansions, namely Gauss-Jackson [2] and Chebyshev-Picard iteration [3,4,5], which have proven to be highly effective. More recently developed techniques are based on spectral collocation methods [6]. This approach discretizes the domain about collocation points, and the solution of the DE is expressed by a sum of "basis" functions with unknown coefficients that are approximated in order to satisfy the DE as closely as possible. Yet, in order to incorporate boundary conditions, one or more equations must be added to enforce the constraints.The Theory of Functional Connections (TFC) is a new technique that analytically derives a constrained expression which satisfies the problem's constraints exactly while maintaining a function that can be freely chosen [7]. This theory, initially called "Theory of Connections", has been renamed for two reasons. First, the "Theory of Connections" already identifies a specific theory in differential geometry, and second, what this theory is actually doing is "functional interpolation", as it provides all functions satisfying a set of constraints in terms of a function ...

Disaster Response Networks (DRNs) are disruption tolerant networks designed to deliver mission critical data during disaster recovery, while operating with limited energy resources. While Quality of Service is desired, it is difficult to offer guarantees because of the unpredictable nature of mobility in such DRNs. The variance of the packet delivery delay (PDV, more commonly called jitter), an important QoS metric which in DRNs is measured in tens of minutes instead of milliseconds, has not been sufficiently addressed in recent research. Smartphones used by first responders generate large data workloads, causing the PDV to further degrade. Reducing packet replication at these workloads will lower energy consumption, but reduces the packet delivery ratio (PDR). The complex interplay between these QoS metrics remains unclear, making their control difficult. We present Raven, a routing protocol for DRNs that offers control over QoS, especially the PDV. Stochastic graph theory which deals with probabilistic edge weights having a mean and variance is used to model mobility in the disaster area. A stochastic version of the K-Shortest Paths algorithm routes data over multiple paths simultaneously. Raven has been thoroughly evaluated in simulation using realistic settings. The dynamics between performance and energy consumption is analyzed mathematically, and its control is demonstrated.