In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the suggested scheme is discussed. This technique is implemented in four examples to illustrate the efficiency and fulfillment of the approach. Examples of numerical solutions to both linear and nonlinear integral equations were used. The Lucas polynomial method and other approaches were contrasted. A collection of tables and figures is used to present the numerical results. We observe that the exact solution differs from the numerical solution if the exact solution is an exponential or trigonometric function, while the numerical solution is the same when the exact solution is a polynomial. The Maple 18 program produced all of the results.

This article deals with a classes of singular integral–differential equations with convolution kernel and reflection. By means of the theory of boundary value problems of analytic functions and the theory of Fourier analysis, such equations can be transformed into Riemann boundary value problems (i.e., Riemann–Hilbert problems) with nodes and reflection. For such problems, we propose a novel method different from classical one, by which the explicit solutions and the conditions of solvability are obtained.

This article discusses a few different types of singular integral equations of the convolution type with Carleman shift in class {0}. By using the theory of Fourier analysis, these equations under consideration are transformed into Riemann–Hilbert boundary value problems for analytic functions with shift and discontinuous coefficients. For such problems, we propose a method different from the classical ones, and we obtain the analytic solutions and the conditions of Noether solvability. MSC2010 Classification: 45E10, 45E05, 30E25