Software for Error Controlled Numerical Solution of Boundary Value Ordinary Differential Equations and Parabolic Partial Differential Equations

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Along with theoretical and experimental science, computational modeling is now viewed as a third fundamental branch of scientific inquiry. A majority of computational models, arising in such areas as chemistry, physics, biology, and finance, involve complex systems of differential equations (DEs), for which approximate solutions are obtained through the use of sophisticated numerical software. Because the numerical solutions are approximate, it is essential that the software also assess the quality of the computed solution. Two popular types of assessment are the global error (the difference between the approximate and exact solutions) and the defect (the amount by which the numerical solution fails to satisfy the equations). Good quality software will adapt a computation so that the numerical solution can be obtained efficiently and so that an estimate of the global error or defect satisfies a user-provided tolerance.