An approach for solving singular two point boundary value problems: analytical and numerical treatment

Abstract

The numerical treatment of two-point singular boundary value problems has always been a difficult and challenging task due to the singularity behaviour that occurs at a point. Various efficient numerical methods have been proposed to deal with such boundary value problems. We present a new efficient modification of the Adomian decomposition method for solving singular boundary value problems, both linear and nonlinear. Numerical examples illustrate the efficiency and accuracy of the proposed method.

Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

“…Usually, we begin by graphically demonstrating the convergence of the approximate solutions Φ n (t) in Equation (19). In Figures 1-3, the approximate solutions Φ 7 (t), Φ 9 (t), and Φ 11 (t) were plotted at a fixed value of λ = 1 for different values of q, where q = 1.5 ( Figure 1), q = 1.6 ( Figure 2), and q = 2 ( Figure 3).…”

“…The ADM was applied to solving algebraic/transcendental/matrix equations [5][6][7][8][9], nonlinear integral/differential equations and both of initial and boundary value problems (IVPs/BVPs), even for irregular boundary contours [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. The solution for this method is an infinite series, which converges when choosing an appropriate canonical form.…”

“…It can be easily checked that the n-term approximate solution for Equation (19) satisfies the initial condition Φ n (0) = λ, ∀n ≥ 1. At t = 0, Equation 19becomes…”

“…= λ. (28) In this section, the Laplace transform and the ADM were applied to obtain the closed form solution for Equation (19) for the Ambartsumian equation. The obtained approximate solutions are investigated in the next section to stand on their domains of applicability and validity.…”

Solution of Ambartsumian Delay Differential Equation with Conformable Derivative

“…However, the conformable derivative (CD) is one of the most prominent operators in this context. To solve the generalized model (Equations (1) and (2)) using the CD, several analytical approaches can be implemented such as the Adomian decomposition method (ADM) [8][9][10][11][12][13][14][15][16][17][18][19][20], the homotopy perturbation method (HPM) [21][22][23], the differential transform method (DTM)/Taylor expansion [24,25], and the the homotopy analysis method (HAM) [6]. In addition, many applications of the CD have been recently discussed by several authors [26][27][28][29].…”