Solution of Extreme Transcendental Differential Equations

Extreme transcendental differential equations are found in many applications including geophysical climate change models. Solution of these systems in continuous time has only been feasible with the recent development of Runge−Kutta sampling transcendental differential equation solvers with Chebyshev function output such as Mathematica 9÷s NDSolve function. This paper presents the challenges and means of solving the widely used DICE 2007 integrated assessment model in continuous time. Application of the solution technique in a mobile policy tool is discussed. Transcendental Differential Equations Transcendental equations contain transcendental or non−algebraic functions such as ex, Log@xD and Cos@xD. The defining characteristic of transcendental functions is that the roots are not algebraically independent, which means that the roots cannot be expressed as the solution to a polynomial equation whose coefficients are polynomials with rational coefficients. DICE 2007 example system It is often difficult to appreciate the sensitivity of complex social systems to changing constraints without simulating key interrelationships in the system. The DICE 2007 integrated assessment model was developed to understand the interrelationships between climate change, the social cost of carbon and efficient carbon abatement trajectories [1], [2], [3]. It has become a classic climate change policy simulation tool for evaluating the social and geophysical effects of global warming. An indicator of the success of models such as DICE 2007 in policy formation is the embedding of results and recommendations within the national climate change policies of many countries. The DICE 2007 model is an optimization problem defined by a system of transcendental differential equations. Figure 1 summarizes a continuous 2007 DICE formulation utilizing parameters fitted through dynamic programming [4], [5]. Maximize utility per capita = 381000 + 1 194 Ù