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Abstract Nonlinear Volterra Equations with Positive Kernels
SIAM Journal on Mathematical Analysis, 1986The author studies the existence of solutions of the equation \[ u(t)+\int^{t}_{0}a(t-s)Au(s)ds\ni f(t),\quad t\geq 0 \] in a Hilbert space. Here the operator A is a nonlinear, possibly multivalued, operator from the space into itself and it is assumed to be pseudomonotone and finitely continuous (for example the sum of a monotone operator defined on ...
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On Nonlinear Volterra Equations with Nonintegrable Kernels
SIAM Journal on Mathematical Analysis, 1980In this paper, the asymptotic behavior of solutions of the nonlinear real Volterra equation \[x(t) + \int_0^t {g(x(t - s))a(s)ds = f(t),\quad t \geqq 0} \] is studied. Here a and f are given and x is the unknown function. The assumptions on the function f are rather weak, and in most cases it is assumed that $\int _0^\infty a(s)ds = + \infty $.
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Fredholm–Volterra integral equation with singular kernel
Applied Mathematics and Computation, 2003The author considers the Fredholm-Volterra integral equation of the second kind \[ \delta\phi(x,t)+\int\limits_{-1}^1 \left| \ln| y-x| -d\right| \phi(y,t)\,dy+\int\limits_0^t F(\tau)\phi(x,\tau) \,d\tau=f(x,t),\tag{1} \] where \(| x| \leq1,\) \( t\in[0,T],\) \(\lambda\in(0,\infty),\) \(\delta\in(0,\infty]\), with a specific right-hand side \(f(x,t ...
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When and why PINNs fail to train: A neural tangent kernel perspective
Journal of Computational Physics, 2022Sifan Wang, Xinling Yu
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