BVP with a Load in the Form of a Fractional Integral

A boundary value problem for a nonhomogeneous heat equation with a load in the form of a fractional Riemann–Liouville integral of an order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>β</a:mi><a:mo>∈</a:mo><a:mfenced open="(" close=")" separators="|"><a:mrow><a:mn>0</a:mn><a:mo>,</a:mo><a:mn>1</a:mn></a:mrow></a:mfenced></a:math> is considered. By inverting the differential part, the problem is reduced to an integral equation with a kernel with a special function. The special function is presented as a generalized hypergeometric function. The limiting cases of the order <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"><f:mi>β</f:mi></f:math> of the fractional derivative are studied: it is shown that the interval for changing the order of the fractional derivative can be expanded to integer values <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"><h:mi>β</h:mi><h:mo>∈</h:mo><h:mfenced open="[" close="]" separators="|"><h:mrow><h:mn>0</h:mn><h:mo>,</h:mo><h:mn>1</h:mn></h:mrow></h:mfenced></h:math> . The results of the study remain unchanged. The kernel of the integral equation is estimated. Conditions for the solvability of the integral equation are obtained.

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