Siddiqi and Akram [23, 24, 25, 26, 27] presented the solutions of 5th, 6th, 8th, 10th, and 12th order boundary value problems using nonpolynomial spline techniques.
Siddiqi and Twizell [18, 19, 20, 21] presented the solutions of 6th, 8th, 10th, and 12th order boundary value problems using 6th, 8th, 10th, and 12th degree splines, respectively.
Although 10 runs were made for each pH solution, only the 1st, 2nd, 3rd, 4th, 5th, and10th runs were analyzed for melamine.
If f : R → R is a C 0 strictly increasing solution of the 4th-order equation (1.3).
(ii) If f : R → R is a C 0 strictly decreasing solution of the 4th-order equation (1.3).
If f : R → R is a C 0 strictly decreasing solution of the 4th-order equation (1.3).
(i) If f : R → R is a C 0 strictly increasing solution of the 4th-order equation (1.3).
By appealing to Sylvester's dialytic elimination method, the direct kinematics problem can be reduced to the solution of a 16th order polynomial equation in a single variable.
On the other hand, assume that f : R → R is a C 0 strictly decreasing solution of the 4th-order equation (1.3).
Then f is a solution given in Theorem 2 of [19]for the characteristic roots r 2, r 3, r 4. (ii) If f : R → R is a C 0 strictly decreasing solution of the 4th-order equation (1.3).
