On the stability of volterra integral equations with seperable kernels

Abstract

Conditions are derived for the uniform stability of a linear scalar Volterra integral equation of the second kind with a seperable (or Pincherle-Goursat) kernel. Certain qualitative behaviour of the solutions of linear systems of differential equations, such a s uniform stability and monotonicity, essential for our analysis, are studied. Use is made of a representation of the resolvent kernel to relate the stability criteria directly in terms of the kernel of the integral equation. Extension of the results to a wider class of integral and integro-differential equations are discussed briefly.

Stability of Collocation Methods for Volterra Integro-Differential Equations

“…We prove that, if the collocation parameters are symmetric, the above methods cannot be A 0 -stable. 1 The stability properties of some collocation methods coincide with those of a suitable associated Runge-Kutta Nyström method applied to the test equation y − λy − μy = 0. However, there are no general results for such methods.…”

“…Before establishing the subsequent result we recall that the following ODE system can be defined as "the ODE system associated to the degenerate VIDE (1.2)" (compare [1,5]):…”

Stability results for one-step discretized collocation methods in the numerical treatment of Volterra integral equations

“…T(t) = (yjk) = (ak(t)bj(t)), j,k=l,...,n, Proof. The condition is sufficient, as it can be proved by a trivial extension of Theorem 3.4 in [1]. It is also necessary; in fact, if the system (3.2) is dissipative with respect to the Euclidean norm, the eigenvalues of ¿(r + T1) are negative or equal to 0.…”

“…We observe that, in all cases, the VDC methods have a stable behavior every time the stability condition is satisfied. However, for problem (C), the numerical solution decreases more slowly than the true solution, and thus the relative error can be large; this is particularly true for the VDC methods (1) and (2). In some cases (method 1 Note that, for method (1) applied to problems (A) and (B), as expected, Xmax is less than 1 if and only if the hypothesis of Theorem 3.3 holds.…”

Reproducing kernel method with Taylor expansion for linear Volterra integro-differential equations

“…Moreover, in [3], there are found mixed interpolation collocation methods to solve first-and second-order Volterra linear integro-differential equations. For methods using a quadrature rule, degenerate kernels, interpolation or extrapolation, homotopy perturbation, Taylor expansion, Chebyshev collocation and Wavelet-Galerkin [4][5][6][7][8][9][10][11]. Recently, the applications of reproducing kernel method (RKM) have become of great interest for scholars [12][13][14][15][16][17][18][19][20][21][22][23][24][25]; In this paper, the use of RKM to solve linear volterra integro-differential equations of the form is introduced.…”