Mapped continuation methods for computing all solutions to general systems of nonlinear equations

• Published in: Computers & chemical engineering Vol. 14; no. 1; pp. 71 - 85
• Main Authors: Seader, J.D., Kuno, M., Lin, W.-J., Johnson, S.A., Unsworth, K., Wiskin, J.W.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 1990; Elsevier
• Subjects: Applications of mathematics to chemical engineering. Modeling. Simulation. Optimization; Applied sciences; Chemical engineering; Exact sciences and technology; Non linear equation; Chemical engineering; Adiabatic; Equation resolution; Continuous stirred tank reactor; Modeling; Steady state
• ISSN: 0098-1354; 1873-4375
• DOI: 10.1016/0098-1354(90)87006-B

In many fields of engineering and science, use of mathematical models leads to systems of linear algebraic and nonlinear algebraic and transcendental equations. When all equations are linear, software packages such as LINPACK and YSMP can be applied to obtain the single solution. When the system contains nonlinear equations, more than one solution may exist, but until recently software packages were designed to obtain at best just one solution from a specified starting guess. If the nonlinear equations are all of polynomial form, recent software packages such as HOMPACK and CONSOL can systematically locate all solutions. The study reported here addresses the general case where the system may contain nonlinear equations with transcendental terms. By forming a fixed-point global homotopy and applying differential arclength continuation in finite mapped space, the two methods described (toroidal mapping and boomerang mapping) have located all solutions from a single starting guess for all cases studied. The two methods are illustrated for the case of an adiabatic continuous stirred-tank reactor operating in a steady-state mode with two consecutive reactions taking place, one of which is catalytic and irreversible with the other noncatalytic and reversible. All five steady-state solutions are found by each method.