Rapid Modeling and Parameter Estimation of Partial Differential Algebraic Equations by a Functional Spreadsheet Paradigm

Chahid Ghaddar

doi:10.3390/mca23030039

Abstract

We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver. Two alternative least-square dynamical minimization schemes are devised and demonstrated on a complex parameterized PDAE system with discontinues properties and coupled time derivatives. Applying the method involves no more than defining a few formulas that closely parallel the original mathematical equations, without any programming skills. It offers a simpler alternative to more complex environments which require nontrivial programming skill and effort.

Highlights

Excel, which is virtually available on every computer, is widely utilized for engineering and scientific applications thanks in part to its simplicity, hundreds of built-in mathematical functions, extensibility, and integrated graphing tools
This paper demonstrates a straightforward method combining the Excel Nonlinear Programming (NLP) solver, which is based on the Generalized Reduced Gradient algorithm [4], with a calculus function encapsulating the Method of Lines (MOL) [5] to carry out the dynamical minimization Equation (1) with only basic Excel spreadsheet skills
The remainder of this paper is organized as follows: we present a formal statement of the partial differential algebraic equations (PDAE) system, as well as a description of the MOL solution strategy

Summary

Introduction

Excel, which is virtually available on every computer, is widely utilized for engineering and scientific applications thanks in part to its simplicity, hundreds of built-in mathematical functions, extensibility, and integrated graphing tools. In Reference [1], we demonstrated how Excel utility could be significantly enhanced to support computational calculus by expanding its built-in math functions with a new breed of calculus functions that accept formulas as a new type of argument. The calculus functions are used in formulas, just like built-in math functions. A calculus integration function accepts a formula and limits as inputs, and computes its integral value—much like an intrinsic math function accepts a number and computes its square root. With appropriate input arguments, a differential system solver function computes and displays a formatted solution in an allocated array much like an intrinsic math function computes and displays an inverse of a matrix. The natural extension of built-in spreadsheet functions allows complete encapsulation of numerical algorithms for solving virtually any computational problem modeled by equivalent formulas using simple input/output pure function evaluation. The calculus functions can be naturally combined with Excel’s built-in Nonlinear Programming (NLP) solver for solving dynamical optimization problems

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