In 1988, Rudolf Lohner developed Fortran-based software for reliable solutions for initial value problems using ordinary differential equations.
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He discovered the necessary and sufficient conditions on initial value problems of ordinary differential equations for the solution to be unique.
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Any differential equation will have many solutions, and each constant represents the unique solution of a well-posed initial value problem.
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If L _ 1 and L _ 2 were constant coefficient matrices, then the exact solution to the associated initial value problem would be
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Do you mean generating N different \ Delta \ mathbf { u } ( 0 ) s and solving an initial value problem with them.
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If we cut out this singularity with a cutoff function, then the so obtained operators still provide solutions to the initial value problem modulo smooth functions.
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Then the above initial / boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space " X ":
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General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at 0 } } with the theta function.
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Then the solution on the interval [ 0, \ tau ] is given by \ psi ( t ) which is the solution to the inhomogeneous initial value problem
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For some " h " " ! where " the " solution to the above equation and initial value problem can be found.
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