Time-dependent and space-dependent source identification problems for partial differential and difference equations take an important place in applied sciences and engineering, and have been studied by several authors. Moreover, the delay appears in complicated systems with logical and computing devices, where certain time for information processing is needed. In the present paper, the time-dependent identification problem for delay hyperbolic equation is investigated. The theorems on the stability estimates for the solution of the time-dependent identification problem for the one dimensional delay hyperbolic differential equation are established. The proofs of these theorems are based on the Dalambert’s formula for the hyperbolic differential equation and integral inequality.

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In the present paper, a time-dependent source identification problem for a one dimensional delay hyperbolic equation with Dirichlet condition is studied. Operator-functions generated by the positive operator are considered. Theorems on the stability estimates for the solution of this problem are established. The first order of accuracy difference scheme for this source identification problem is presented. Numerical analysis and discussions are presented.

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