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In many cases, analytic solutions of partial differential equations may notbe possible. For practical problems, it is more reasonable to carry outcomputational solutions. However, the standard grid in the finite differenceapproximation is not a correct model of the continuous domain in terms ofdigital topology. In order to avoid serious problems in computational solutionsit is necessary to use topologically correct digital domains. This paperstudies the structure of the hyperbolic partial differential equation on graphsand digital n-dimensional manifolds, which are digital models of continuousn-manifolds. Conditions for the existence of solutions are determined andinvestigated. Numerical solutions of the equation on graphs and digitaln-manifolds are presented.

Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: a Klein Bottle a Projective Plane and a 4D Sphere