An approximate method for solving a class of weakly-singular Volterra integro-differential equations

Solving nonlocal boundary value problems for first‐ and second‐order differential equations by the Adomian decomposition method

“…Consider the nonlocal BVP for the Euler equation: Equation 59 Proceeding as in Example 4, substituting equations (3.25)‐(3.27) into this nonlocal BVP, where φ ( x )= φ 1 ( x )+ φ 2 ( x )= k , we obtain: Equation 60 or: Equation 61 Due to the singular behavior of the functions in the neighborhood of the point x =0, the solution cannot be obtained by the usual decomposition method. Then, as in Bougoffa et al (2011) and Bougoffa et al (2012) we can add a correction term v ″ to both sides. Thus: Equation 62 Therefore, we set the recursion scheme: Equation 63 where Rv =− x 2 v ‴+(1− x ) v ″+ v ′.…”

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

“…Numerical solution of an integro‐differential equation of parabolic type with a memory term containing a weakly singular kernel by using means of the Galerkin finite element method has been presented in Chen et al A spectral Jacobi‐collocation approximation for the linear Volterra integral equations of the second kind with weakly singular kernels has been analyzed in Chen and Tang . A new approach to resolve linear and nonlinear weakly‐singular Volterra integro‐differential equations of first‐ or second‐order has been developed in Bougoffa et al In this method, the singularity is first removed using Taylor's approximation and then integro‐differential equation is transformed into an ordinary differential equation. The numerical solution of a cauchy‐type singular integral equation of the first kind, which occurs rather frequently in mathematical physics, has been presented in Dezhbord et al This numerical scheme is based on reproducing the kernel Hilbert space method, which offers the solution in the form of a series in the reproducing kernel space.…”

Solving Cauchy integral equations of the first kind by the Adomian decomposition method

“…Special integral equations such as weakly-singular Volterra integro-differential equations, Abel integral equations and some types of linear and nonlinear integral equations of the first kind have been transformed into a canonical form so that the Adomian decomposition method can be applied, which permits convenient resolution of these equations [19][20][21][22][23].…”