Generalized solution of mixed value problem for a linear integro-differential equation with pseudoparabolic operator of higher power

T. K. Yuldashev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 4, pp. 34-43.

Résumé

Mathematical modeling of many processes occurring in the real world leads to the study of initial and boundary value problems for equations of mathematical physics. Mixed value problems for partial differential and integro-differential equations by virtue of their importance in the application are one of the most important parts of the theory of differential equations. We propose a method of studying the one-value generalized solvability of the mixed value problem for a linear higher-order pseudoparabolic type of integro-differential equation with degenerate kernel. Integro-differential equations of such type model many natural phenomena and appear in many fields of sciences. For this reason, this type of equations was given a great importance in the works of many researchers. In this article in rectangular domain D we consider the questions of solvability and constructing the generalized solution of mixed value problem for a linear integro-differential equation with pseudoparabolic operator of higher power and degenerate kernel $$ \mathfrak{J}^{n} U(t, x) = \mu \int_{0}^{T}K(t,s)U(s,x) ds+\alpha(t)U(t,x) (1)$$ with initial $$U(t,x)_{|t=0}=\varphi_{1}(x), \dfrac{\partial^{j-1}}{\partial t^{j-1}}U(t,x)_{|t=0}=\varphi_{j}(x), j=\overline{2, n} (2)$$ and Benar-type boundary value conditions $$U(t,x)_{|x=0}=U_{xx}(t,x)_{|x=0}=\ldots=\dfrac{\partial^{2(2nm-1)}}{\partial x^{2(2nm-1)}}U(t,x)_{|x=0}=$$ $$=U(t,x)_{|x=l}=U_{xx}(t,x)_{|x=l}=\ldots=\dfrac{\partial^{2(2nm-1)}}{\partial x^{2(2nm-1)}}U(t,x)_{|x=l}=0 (3)$$ where $$f (x,u)\in C(D_{l}\times R), \varphi_{j}(x)\in C^{4mn+1}(D_{l}), \varphi_{j}(x)_{|x=0}=\varphi_{j}^{\prime\prime}(x)_{|x=0}= $$ $$ =\ldots=\varphi_{j}^{4nm-2}(x)_{x=0}=\varphi_{j}(x)_{|x=l}=\varphi_{j}^{\prime\prime}(x)_{|x=l}=\ldots=\varphi_{j}^{4nm-2}(x)_{x=l}=0, j=\overline{1, n},$$ $$ K(t,s)=\sum_{i=1}^{k}a_{i}(t)b_{i}(s), a_i(t), b_i(s)\in C^n(D_T), \alpha(t)\in C^n(D_T), 0\nu $$ is small parameter, $\mu$ is real spectral parameter, $D\equiv D_T\times D_l, D_T\equiv[0,T], D_l\equiv[0,l], 0$ and $m$ are fixed natural numbers and $$\mathfrak{J}^{n}=\left( \dfrac{\partial}{\partial t}+(-1)^m \nu\dfrac{\partial^{2m+1}}{\partial t\partial x^{2m}}+\dfrac{\partial^{4m+1}}{\partial t\partial x^{4m}}+\dfrac{\partial^{4m}}{\partial x^{4m}}\right)^n. $$ Here we suppose that the functions $a_i(t)$ and $b_i(s)$ are linear independent. We use the method of Fourier series based on separation of variables. Application of this method of separation of variables can improve the quality of formulation of the considering mixed value problem and facilitates the processing procedure. With the introduction of the notation we obtained the system of countable system of algebraic equations. From the condition of non-degenerate of Fredholm determinant we calculate the regular values of parameter $\nu$. Solving this algebraic system for these regular values of parameter $\nu$ we can reduce consideration of the mixed value problem to the countable system of linear integral equation, one-value solvability of which is proved by the method of successive approximation. The criterion of one-value solvability of the considered problem is established. Under this criterion we prove the theorems of one-valued generalized solvability of the mixed value problems. Every estimate was obtained by the aid of the Hölder inequality and Minkovski inequality. This paper advances the theory of partial integro-differential equations with degenerate kernel.

Détail

Keywords: mixed value problem, linear integro-differential equation, degenerate kernel, spectral parameter, slightly generalized solvability.

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