Fredholm-Volterra integral equation with singular kernel.
The 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 \(f(x,t)\), under the condition \[ \int\limits_{-1}^1 \phi(y,t)\,dy=P(t),\qquad \int\limits_{-1}^1y\phi(y,t)\,dy=M(t). \] The equation (1) arises in a contact problem in elasticity. Using the quadrature formula for the second integral, a system of integral equations is obtained with respect to the unknowns \(\phi_k(x)=\phi(x,t_k)\), \(k=1,\ldots,N\). Using the Legendre polynomials the obtained system with the logarithmic kernel is further reduced to an infinite algebraic system. The case \(\delta=0\) (a first kind equation) is considered separately. In conclusion a numerical result is demonstrated.
scientific article; zbMATH DE number 1977955
scientific article