Abstract
In this paper, we propose two generalized non-polynomial cubic spline schemes using a variable mesh to solve the system of non-linear singular two point boundary value problems. Theoretical analysis proves that the proposed methods have second- and third-order convergence. Both methods are applicable to singular boundary value problems. Numerical results are also provided to show the accuracy and efficiency of the proposed methods.
Highlights
1 Introduction In this paper, we study two effective numerical techniques using a non-polynomial cubic spline based on a variable mesh to solve system of M non-linear singular boundary value problems (BVPs) of the following type: y(xix) = F(i) x, y( ), . . . , y(i), . . . , y(M), y(x ), . . . , y(xi), . . . , y(xM), a ≤ x ≤ b, ( )
In Section, we present the application of the proposed schemes to a fourth- and sixth-order singular BVP
The linear system of difference equations have been solved by the block Gauss elimination method and the non-linear system of difference equations by the block Newton’s method in which we have considered y = as the initial approximation
Summary
We have derived generalized non-polynomial cubic spline schemes of second- and third-order using a variable mesh for solving system of two point boundary value problems ( )-( ). Applying the difference scheme ( ) to the coupled second-order boundary value problem ( )-( ), we obtain the following difference scheme: σjyj– – ( + σj)yj + yj+ = hjhj+ (Pjzj+ + Qjzj + Rjzj– ), σjzj– – ( + σj)zj + zj+ = hjhj+ Pj(aj+ zxj+ + bj+ zj+ + cj+ yxj+ + dj+ yj+ + gj+ ). Theorem The scheme ( ) for the numerical solution of system of non-linear singular boundary value problem ( ) with sufficiently small hj and σ = has third-order convergence under appropriate conditions
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