The numerical task is made difficult by the dimensionality and geometry of the independent variables, the nonlinearities in the system, sensitivity to boundary conditions, a lack of formal understanding of the kind of solution method to be employed for a particular problem, and so on.Ī number of methods have been developed to deal with the numerical solution of PDEs.
However, few PDEs have closed-form analytical solutions, making numerical methods necessary. Examples range from the simple (but very common) diffusion equation, through the wave and Laplace equations, to the nonlinear equations of fluid mechanics, elasticity, and chaos theory. Mathematical problems described by partial differential equations (PDEs) are ubiquitous in science and engineering. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations.
MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. The maximum relative curvature diagnostics are useful if you wish to make inferences based on the linear approximation to the nonlinear model.A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. However, now the asymptotic correlation between parameters is high for all pairs, indicating that a model with fewer parameters should be considered. This indicates that the least-squares estimates of the parameters, , and have nearly linear behavior. However, here reparametrization has reduced the parameter-effects curvature below the critical value of 0.22843. Note that reparametrizing the model does not affect the intrinsic curvature. ParameterBias is based on the average curvature of the solution locus tangential to the least-squares estimate. If the maximum relative parameter-effects curvature is small compared to the confidence region relative curvature, the parameter coordinates projected onto the tangential plane are approximately parallel and uniformly spaced over the confidence region. If the maximum relative intrinsic curvature is small compared to the confidence region relative curvature, the solution locus is approximately planar over the confidence region. Similarly, maximizing the relative parameter-effects curvature over the p-dimensional subspace tangential to the locus gives the maximum relative parameter-effects curvature.īoth of these quantities can be compared to the relative curvature of the confidence region centered on the least-squares parameter estimates. Maximizing the relative intrinsic curvature over the ( n- p )-dimensional subspace normal to the locus gives the maximum relative intrinsic curvature. Standardizing curvature to be response-invariant gives relative curvature. Parameter-effects curvature describes the tangential component of the solution locus curvature at the least-squares estimate. Intrinsic curvature describes the normal component of the solution locus curvature at the least-squares estimate. NonlinearRegress įind numerical values of the parameters pars that make the model expr give a best fit to data as a function of vars and provide diagnostics for the fitting