Inverse Problem for Determining Time-Dependent Coefficient and Source Functions in a Time-Fractional Telegraph Equation

This work focuses on the inverse problem of identifying time-dependent coefficients in a time-fractional telegraph equation. The governing equation under study is given by $$\left(D_{0t}^{\alpha}\circ D_{0t}^{\alpha}\right)u(x,t)+2aD_{0t}^{\alpha}u(x,t)-u_{xx}(x,t)+r_{1}(t)u(x,t)=r_{2}(t)f(x,t),$$ where $$0<t\leq T$$ , $$0\leq x\leq\pi$$ , $$0<\alpha<1$$ , and $$D_{0t}^{\alpha}$$ denotes the Caputo fractional derivative. We first investigate the Cauchy problem. By the separating variables method, the Cauchy problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and generalized singular Gronwall inequalities, an estimate for the solution of the Cauchy problem is obtained in terms of the norm of the unknown functions. The inverse problem is reduced to the equivalent system of integral equations. For solving this system, the contracted mapping principle is applied.

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