Abstract
In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.
Highlights
The Theory of Connections (ToC), introduced by D
Linear or iterative non-linear least-squares is used to solve for the coefficients of this expansion. This approach to solve ordinary differential equations (ODEs)/partial differential equations (PDEs) has many advantages over traditional methods: (1) it consists of a unified framework to solve IVP, BVP, or multi-values problems, (2) it provides an approximated solution expressed via analytical functions that can be used for subsequent algebraic manipulation, (3) the solution accuracy is usually obtained fast for many application problems, (4) the procedure can be numerically robust, and (5) it can solve ODE/PDE subject to a variety of constraint types: absolute, relative, linear, non-linear, and integral
The purpose of this paper is to introduce a set of five new applications of Theory of Connections; applications that are not covered in the previous references and where ToC is found effective
Summary
The Theory of Connections (ToC), introduced by D. Linear or iterative non-linear least-squares is used to solve for the coefficients of this expansion This approach to solve ODEs/PDEs has many advantages over traditional methods: (1) it consists of a unified framework to solve IVP, BVP, or multi-values problems, (2) it provides an approximated solution expressed via analytical functions that can be used for subsequent algebraic manipulation, (3) the solution accuracy is usually obtained fast for many application problems, (4) the procedure can be numerically robust (very low condition number), and (5) it can solve ODE/PDE subject to a variety of constraint types: absolute, relative, linear, non-linear, and integral. This extension represents the multivariate formulation of ToC subject to arbitrary-order derivative constraints in rectangular domains This provides an analytical procedure to obtain constrained expressions in any orthogonal/rectangular space that can be used to transform constrained problems into unconstrained problems. The complete solution derivations of each one of these new applications, validated by numerical examples, are detailed in the subsequent sections
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