# On a problem for the parabolic-hyperbolic type equation of fractional order with non-linear loaded term

**Authors:**Abdullaev O.K.^{1}-
**Affiliations:**- V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science,

**Issue:**Vol 25, No 1 (2021)**Pages:**7-20**Section:**Differential Equations and Mathematical Physics**URL:**https://journals.eco-vector.com/1991-8615/article/view/60083**DOI:**https://doi.org/10.14498/vsgtu1777**ID:**60083

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## Full Text

## Abstract

We study the existence and uniqueness of solution of the non-local problem for the parabolic-hyperbolic type equation with non linear loaded term involving Caputo derivative

\[

f(x) = \begin{cases}

{u_{xx}}-_CD_{0y}^\alpha u+a_1(x)u^{p_1}(x,0), & y > 0,

\\

{u_{xx}}-{u_{yy}}+a_2(x)u^{p_2}(x,0), & y<0,

\\

\end{cases}

\]

where

\[

{}_CD_{0y}^{\alpha }f(y) = \frac{1}{{\Gamma (1-\alpha )}}\int_0^y {(y - t)}^{-\alpha}f'(t)\,dt, \quad 0 < \alpha < 1,

\]

\(a_i(x)\) are given functions, \(p_i\), \(\alpha=\mathrm{const}\), besides \(p_i>0\) \((i=1,2)\), \(0 < \alpha < 1\) in the domain \(\Omega\) bounded with segments:

\[

A_1 A_2 = \{ (x,y): x = 1, 0 < y < h\},\quad B_1 B_2 = \{ (x,y): x = 0, 0 < y < h\},

\]

\[

B_2 A_2 = \{ (x,y): y = h, 0 < x < 1\}

\]

at the \(y > 0\), and characteristics:

\[

A_1C: x - y = 1,\quad B_1C: x + y = 0

\]

of the considered equation at \(y < 0\), where \(A_1 (1, 0)\), \(A_2 (1, h)\), \(B_1( 0, 0)\), \(B_2( 0, h)\), and \(C(1/2, -1/2)\).

Uniqueness of solution of the investigated problem was proved by an integral of energy. The existence of solution of the problem was proved by the method of integral equations. The theory of the second kind Fredholm type integral equations and the successive approximations method were widely used. We notice, that boundary value problems for the mixed type equations of fractional order with non linear loaded term have not been investigated.

## About the authors

### Obidjon Kh. Abdullaev

V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science,
**Author for correspondence.**

Email: obidjon.mth@gmail.com

ORCID iD: 0000-0001-8503-1268

SPIN-code: 6441-5987

Scopus Author ID: 57190022167

ResearcherId: AAJ-1572-2020

http://www.mathnet.ru/rus/person66335

Cand. Phys. & Math. Sci.; Associate Professor; Doctoral Student; Dept. of Differential Equations and Their Application

4-a, Universitetskaya st., Tashkent, 100174, Uzbekistan## References

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