Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

Abstract

A new algorithm for eigenvalue problems for the fractional Jacobi type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of m terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides an super-exponential convergence rate as m → ∞. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for m = ∞ and a truncated symbolic algorithm (TSA) for a finite m. Numerical examples are presented to support the theory.

Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients

“…The sufficient conditions of an exponential convergence rate and the absolute errors estimates of the proposed approach are received in Section 5. The obtained absolute errors estimates of the FD-method significantly improve the accuracy of the estimates obtained earlier in [5]. Derivation of the basic formulas for the proposed new symbolic algorithmic implementation of our method is given in Section 6 and in Appendices A, B, C. The numerical algorithm is given in Section 7.…”

“…In [5] it was shown that the fourth order ordinary differential equation with all derivatives of the eigenfunction could be reduced to the form (1) using the variable transformation. That is why we consider the eigenvalue problem with equation (1).…”

“…The analytical methods based on perturbation and homotopy ideas [1,2] are widely used for solving the eigenvalue problems. The numerical-analytical (functional-discrete) methods refer to these methods (see, for example, [3,4,5,6,7,8,9,10] and compare with Adomian decomposition method [11,12]). Using these analytical methods the solutions can be found as fast convergent functional series.…”

“…the approach can be applied also to eigenvalue problems with multiple eigenvalues (see, for example, [10]); 3. all eigenpairs can be computed in parallel; 4. the convergence rate increases as the index of the eigenpair increases; 5. it was proved that in many cases the FD-method converges exponentially or super-exponentially.…”

“…Briefly described general idea was used for developing and justification of new symbolic algorithms for FD-method for the Sturm-Liouville problems on a finite interval for the Schrödinger equation with a polynomial potential in [6,8]. Using the described idea for the fourth order Sturm-Liouville problem, in this article we modify the traditional method from [5,10] and develop a new symbolic algorithm of the FD-method. The proposed algorithm of our method is developed when the potential coefficients are approximated by zero function.…”