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Boundary value problems for partial difference equations

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The generalized boundary value problem for a linear multidimensional (partial) difference equation with nonconstant coefficients on an arbitrary set in the multidimensional discrete space is formulated. For the equation with a dominant (or semidominant) central coefficient, theorems concerning the uniqueness of the solution are proved (Section 2). In Section 3, the definition and known general properties of the fundamental solution are recalled, in particular if the coefficients of the equation are constants. The fundamental solution is used for the proof of existence of the solution as well as for its construction. This problem is dealt with in the remaining parts of the paper. Section 4. is devoted to the case when the fundamental solution tends to zero limit at infinity and Section 5. to the case when the fundamental solution logarithmically increases.

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This work has been supported by the Grant No. 8188 of the Czech Technical University, Prague and by the Grant No. 201/93/0932 of the GA CR.

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Veit, J. Boundary value problems for partial difference equations. Multidim Syst Sign Process 7, 113–134 (1996). https://doi.org/10.1007/BF01827809

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Key words

  • linear difference equations
  • n-D sequences
  • boundary value problems
  • fundamental solution
  • bounded solutions
  • generalized boundary conditions