The Neumann problem is used to model many linear and nonlinear phenomena such as electrostatic problems, acoustic problems, vibrations of a string, fluid flow problems, the evolution of an isolated population, etc. This paper proposes a numerical technique to solve second-order linear partial differential equations with variable coefficients subject to the Neumann boundary condition (i.e., the boundary condition of the second kind). Our technique uses the operational matrix method and standard collocation points and approximates the solution using Lerch polynomials bases. Also, we enhance the method's effectiveness by utilizing an error analysis technique based on residual function. The implementation of our method to any computer program is more straightforward than many other numerical methods. The results of numerical experiments are illustrated with tables and figures and are compared with analytical solutions to confirm the good accuracy of the presented technique. KeywordsLerch series • Matrix and collocation methods • Neumann boundary conditions • Partial differential equations • Residual error analysis B Seda Çayan

In this study, second order linear Volterra partial integro-differential equation with two-and three-dimensional are solved by collocation method based on Lerch polynomials. This method is composed of the operational matrix and collocation methods, which are based upon the matrix forms of the Lerch polynomials with the parameter λ and Taylor polynomials, and their derivatives and integrals. The approximate solutions of the mentioned equations are investigated in terms of the Lerch polynomials the different values of λ. Also, to verify the accuracy and efficiency of the present method, an alternative convergence criterion along with error analysis depending on residual function is enhanced. Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.

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Curves of constant width, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, a system of differential equations characterizing the curves of constant width is examined. This is the system of the first order homogenous differential equations with variable coefficients in the normal form. Approximate solutions of the system, by means of two different polynomial approaches, are calculated and error analysis is made. The obtained results are analyzed on a numerical sample and the best method of approach is determined. This system can also constitute a characterization for different types of curves according to different frames in different spaces. Therefore, this study is important not only for this curve type but also for the geometry of all curves that can be expressed in a similar system.