In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian pol...

Truncated Power Series technique (Differential Algebra or DA) is a powerful tool for non-linear map analysis of accelerators. The most natural language for numerical DA’s is C++, since it is object oriented and has operator overloading. Traditional C++, though, can be inefficient for scientific programming due to creation of many temporaries and extra loops in overloaded operators. Recent Expre...

Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in Rn. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or “interpolated”. The sequence of univariate calculations is more efficient than multivariat...

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations generalized Caputo derivatives that appear in the study astronomical objects. The offered solution is approximated as truncated series normalized shifted Jacobi polynomials under assumption exact an element L2. method used solver obtain unknow...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...