I have attached there some equations which is needed to be solved by keller box method.but I have faced problems with block elimination portion because of here momentum equation starts with f'' instead of f'''.I have also attached here sample matrix when equation starts with f'''.what will happen when it starts with f''?what will be iteration of converges for this?
From the transformed boundary conditions it appears that derivatives in your equations are with respect to η and that all other quantities in these equations are constants. If so, your 1st, 3rd and 4th differential equations are linear ordinary differential equations, which can be immediately integrated and expressed in terms of other functions. Only your 2nd equation is a non-linear differential equation in need of special treatment. To clarify, express your first differential equation as:
(1) f ' ' (η) = Φ(η),
where
(2) Φ(η) := R ax (θ'(η) - Ν r φ'(η) - R b ς'(η))/(P ex).
in which I assume θ'(η), φ'(η) and ς'(η) are well-specified functions of η. The most general solution of the differential equation in Eq. (1) is
(3) f (η) = Α η + Β + ∫0η dη' (η-η') Φ(η'),
where A and B are constants, independent of η. From the condition f (η=0) = 0 it follows that B = 0. From f ' (η → ∞) ~ 0 it further follows that A = 0, and in addition that
(4) ∫0∞ dη' Φ(η') = 0,
where I have used the first integral of the differential equation in Eq. (1), namely
(5) f ' (η) = A + ∫0η dη' Φ(η').
From the defining equation in Eq. (2) one has
(6) ∫0∞ dη' Φ(η')
= R ax {[θ(∞) - Ν r φ(∞) - R b ς(∞)] - [θ(0) - Ν r φ(0) - R b ς(0)]}/(P ex)
≡ - R ax (1 - N r - R b)/(P ex),
where in the last equality I have used the boundary conditions specified by you. From Eqs (4) and (6) one arrives at the condition
(7) N r + R b = 1.
Thus, of the four parameters N, R, r and b only three can be chosen independently.
You can treat your 3rd and 4th equations along the same lines as above. It remains only to treat your 2nd equation by means of an appropriate numerical method, such as the one by Atkinson, or Keller. As regards Keller's method, I recommend you to consult his extensive discussion of his method in the book Applications of Bifurcation Theory, edited by PH Rabinowitz (Academic Press, New York, 1977), pp. 359-384 (after studying this paper, you will realise that the 'box' in 'the Keller box' is nothing extraordinary). As regards Atkinson's method, see KE Atkinson, The Numerical Solution of A Bifurcation Problem, SIAM J. Numer. Anal. 14, 584 (1977). □
From the transformed boundary conditions it appears that derivatives in your equations are with respect to η and that all other quantities in these equations are constants. If so, your 1st, 3rd and 4th differential equations are linear ordinary differential equations, which can be immediately integrated and expressed in terms of other functions. Only your 2nd equation is a non-linear differential equation in need of special treatment. To clarify, express your first differential equation as:
(1) f ' ' (η) = Φ(η),
where
(2) Φ(η) := R ax (θ'(η) - Ν r φ'(η) - R b ς'(η))/(P ex).
in which I assume θ'(η), φ'(η) and ς'(η) are well-specified functions of η. The most general solution of the differential equation in Eq. (1) is
(3) f (η) = Α η + Β + ∫0η dη' (η-η') Φ(η'),
where A and B are constants, independent of η. From the condition f (η=0) = 0 it follows that B = 0. From f ' (η → ∞) ~ 0 it further follows that A = 0, and in addition that
(4) ∫0∞ dη' Φ(η') = 0,
where I have used the first integral of the differential equation in Eq. (1), namely
(5) f ' (η) = A + ∫0η dη' Φ(η').
From the defining equation in Eq. (2) one has
(6) ∫0∞ dη' Φ(η')
= R ax {[θ(∞) - Ν r φ(∞) - R b ς(∞)] - [θ(0) - Ν r φ(0) - R b ς(0)]}/(P ex)
≡ - R ax (1 - N r - R b)/(P ex),
where in the last equality I have used the boundary conditions specified by you. From Eqs (4) and (6) one arrives at the condition
(7) N r + R b = 1.
Thus, of the four parameters N, R, r and b only three can be chosen independently.
You can treat your 3rd and 4th equations along the same lines as above. It remains only to treat your 2nd equation by means of an appropriate numerical method, such as the one by Atkinson, or Keller. As regards Keller's method, I recommend you to consult his extensive discussion of his method in the book Applications of Bifurcation Theory, edited by PH Rabinowitz (Academic Press, New York, 1977), pp. 359-384 (after studying this paper, you will realise that the 'box' in 'the Keller box' is nothing extraordinary). As regards Atkinson's method, see KE Atkinson, The Numerical Solution of A Bifurcation Problem, SIAM J. Numer. Anal. 14, 584 (1977). □
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