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Boundary value problem for mixed-compound equation with fractional derivative, functional delay and advance. (Russian. English summary) Zbl 1438.35420

Summary: We study the Tricomi problem for the functional-differential mixed-compound equation \(LQu(x,y)=0\) in the class of twice continuously differentiable solutions. Here \(L\) is a differential-difference operator of mixed parabolic-elliptic type with Riemann-Liouville fractional derivative and linear shift by \(y\). The \(Q\) operator includes multiple functional delays and advances \(a_1(x)\) and \(a_2(x)\) by \(x\). The functional shifts \(a_1(x)\) and \(a_2(x)\) are the orientation preserving mutually inverse diffeomorphisms. The integration domain is \(D=D^+\cup D^-\cup I\). The “parabolicity” domain \(D^+\) is the set of \((x,y)\) such that \(x_0<x<x_3\), \(y>0\). The ellipticity domain is \(D^-=D_0^-\cup D_1^-\cup D_2^-\), where \(D_k^-\) is the set of \((x,y)\) such that \(x_k<x<x_{k+1}\), \(-\rho_k(x)<y<0\), and \(\rho_k=\sqrt{a_1^k(x)(x_1-a_1^k(x))}\), \(\rho_k(x)=\rho_0(a_1^k(x))\), \(k=0, 1, 2\). A general solution to this Tricomi problem is found. The uniqueness and existence theorems are proved.

MSC:

35R10 Partial functional-differential equations
35M13 Initial-boundary value problems for PDEs of mixed type
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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References:

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