arXiv subjects

Rajapboy Saparbayev

Publications and source records attributed to Rajapboy Saparbayev.

3 recordsLinked to original sources

arXiv2023

Fractional Telegraph equation with the Riemann-Liouville derivative

The Telegraph equation $(\partial_{t}^{ρ})^{2}u(x,t)+2α\partial_{t}^{ρ}u(x,t)-u_{xx}(x,t)=f(x,t)$, where $0<t\leq T$ and $0<ρ<1$, with the Riemann-Liouville derivative is considered. Existence and uniqueness theorem for the solution to the problem under consideration is proved. Inequalities of stability are obtained. The applied method allows us to study a similar problem by taking instead of $d^2/dx^2$ an arbitrary elliptic differential operator $A(x, D)$, having a compact inverse.

math.AP↗

arXiv2023

Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form $(D_{t}^ρ)^{2}u(t)+2αD_{t}^ρu(t)+Au(t)=p( t)q+f(t)$, where $0<t\leq T$, $0<ρ<1$ and $D_{t}^ρ$ is the Caputo derivative. The equation contains a self-adjoint positive operator $A$ and a time-varying multiplier $p(t)$ in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition $B[u(t)] = ψ(t)$ is imposed, where $B$ is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator $A$ and the functional $B$ are discussed.

math.AP↗

arXiv2023

Fractional Telegraph equation with the Caputo derivative

The Cauchy problem for the telegraph equation $(D_{t}^{ρ})^{2}u(t)+2αD_{t}^{ρ}u(t)+Au(t)=f(t)$ ($0<t\leq T, \, 0<ρ<1$), with the Caputo derivative is considered. Here $A$ is a selfadjoint positive operator, acting in a Hilbert space $H$, $D_t$ is the Caputo fractional derivative. Existence and uniqueness theorems for the solution to the problem under consideration is proved. Inequalities of stability are obtained.

math.AP↗