Abstract
In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority of the proposed method.
Highlights
Spectral approaches are a class of schemes used in applied mathematics and scientific computing to numerically solve certain differential equations and nonlinear integral equations
These approaches have been used in modeling of many problems of physical phenomena, engineering and chemical processes in chemical kinetics [1], super fluidity biology and economics [2], axially symmetric problems in the case of an elastic body containing an inclusion [3], and fluid dynamics [4], and the Hammerstein integral equation is employed for modeling nonlinear physical phenomena such as electromagnetic fluid dynamics reformulation of boundary value problems with a nonlinear boundary condition [5]
Genocchi polynomials and Genocchi numbers have been widely applied in many branches of mathematics and physics such as complex analytic number theory, homotopy theory, differential topology, and quantum physics [25,26]
Summary
Spectral schemes are invaluable tools for the numerical solution of fractional partial differential equations (FPDEs), ordinary differential equations (ODEs), integral equations (IEs), and integrodifferential equations (IDEs). Spectral approaches are a class of schemes used in applied mathematics and scientific computing to numerically solve certain differential equations and nonlinear integral equations. Various numerical approaches have been presented for solving a class of nonlinear singular integral equations including Abel’s integral equation, Hammerstein integral equation, Volterra integral equation, etc. Presented a spectral method based on the second Chebyshev wavelet (SCW) operational matrix for solving the fractional nonlinear Fredholm integrodifferential equation, and the Ferdholm and Volterra integral equations. Nemati in [15] applied a numerical approach for solving nonlinear fractional integrodifferential equations with weakly singular kernels by using a modification of hat functions. We use efficient functions such as Genocchi polynomials and their operational matrices to solve nonlinear Volterra integral equations with weakly singular kernels of the following form: t.
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