Abstract

In this paper, we consider an impulsive homogeneous parabolic type partial integro-differential equation with degenerate kernel and involution. With respect to spatial variable \(x\) is used Dirichlet boundary value conditions and spectral problem is studied. The Fourier method of separation of variables is applied. The countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of countable system of functional equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The unique solution of the impulsive mixed problem is obtained in the form of Fourier series. Absolutely and uniformly convergence of Fourier series is proved.

中文翻译：

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具有求和和非线性条件的脉冲抛物型积分微分方程的混合问题

 抽象的

在本文中，我们考虑具有简并核和对合的脉冲齐次抛物型偏积分微分方程。针对空间变​​量\(x\)，采用Dirichlet边值条件，研究谱问题。应用变量分离的傅里叶方法。针对未知函数的傅里叶系数，得到了可数非线性函数方程组。证明了可数函数方程组唯一可解性定理。逐次逼近法与收缩映射法结合使用。脉冲混合问题的唯一解以傅里叶级数的形式得到。证明了傅里叶级数的绝对一致收敛性。