Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation

ABSTRACT

Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.

KEYWORDS

Interval Functions, Interval Majorant, Interval Extension, Interval Operator, Nested Sequence

Interval Functions, Interval Majorant, Interval Extension, Interval Operator, Nested Sequence

Cite this paper

Arawomo, P. (2014) Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation.*Advances in Pure Mathematics*, **4**, 147-155. doi: 10.4236/apm.2014.44020.

Arawomo, P. (2014) Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation.

References

[1] Lakshmikantham, V. and Pandit, S.G. (1985) The Method of Upper and Lower Solution and Hyperbolic Partial Differential Equations. Journal of Mathematical Analysis and Applications, 105, 466-477.

http://dx.doi.org/10.1016/0022-247X(85)90062-9

[2] Moore, R.E. (1979) Methods and Application of Interval Analysis. SIAM (Studies in Applied and Numerical Mathematics), Philadelphia. http://dx.doi.org/10.1137/1.9781611970906

[3] Arawomo, P.O. and Akinyele, O. (2002) An Interval Analytic Method in Constructive Existence Result for Initial Value Problems. Dynamic Systems and Applications, 11, 545-556.

[4] Chan, C.Y. and Vatsala, A.S. (1990) Method of Upper and Lower Solution and Interval Method for Semilinear Euler-Pision-Darboux Equations. Journal of Mathematical Analysis and Applications, 150, 378-381.

http://dx.doi.org/10.1016/0022-247X(90)90110-2

[5] Lakshmikantham, V. and Swansundaran, S. (1987) Interval Method for 1st Order Differential Equations. Applied Mathematics and Computation, 23, 1-5. http://dx.doi.org/10.1016/0096-3003(87)90052-X

[6] Moore, R.E. (1984) Upper and Lower Bounds on Solutions of Differential, Integral Equations without Using Monotonicity, Convexity or Maximum Principle., In: Vichnevetsky, R. and Stepheman, R., Eds., Advances in Computer Methods for Partial Differential Equations, IMACS, 458-461.

[7] Moore, R.E. (1984) A Survey of Interval Methods for Differential Equations. Proceedings of 23rd Conference on Decision and Control, Las Vegas, 12-14 December 1984, 1529-1535.

[8] Walter, W. (1970) Differential and Integral Inequalities. Springer-Verlag, Berlin, Heidelberg.

http://dx.doi.org/10.1007/978-3-642-86405-6

[9] Caprani, O. and Madsen, K. (1975) Contraction Mappings in Interval Analysis. BIT Numerical Mathematics, 15, 362-366. http://dx.doi.org/10.1007/BF01931673

[10] Moore, R.E. (1977) A Test for Existence of Solution to Nonlinear Systems. SIAM Journal on Numerical Analysis, 14, 611-615.

[11] Moore, R.E. (1978) Bounding Sets in Function Spaces with Application to Nonlinear Operator Equations. SIAM Review, 20, 492-512.

[12] Rall, L.B. (1983) Mean-Value and Taylor Forms in Interval Analysis. SIAM Journal on Mathematical Analysis, 14, 223-238.

[13] Rall, L.B. (1982) Integration of Interval Functions II, the Finite Case. SIAM Journal on Mathematical Analysis, 13, 690-698.

[1] Lakshmikantham, V. and Pandit, S.G. (1985) The Method of Upper and Lower Solution and Hyperbolic Partial Differential Equations. Journal of Mathematical Analysis and Applications, 105, 466-477.

http://dx.doi.org/10.1016/0022-247X(85)90062-9

[2] Moore, R.E. (1979) Methods and Application of Interval Analysis. SIAM (Studies in Applied and Numerical Mathematics), Philadelphia. http://dx.doi.org/10.1137/1.9781611970906

[3] Arawomo, P.O. and Akinyele, O. (2002) An Interval Analytic Method in Constructive Existence Result for Initial Value Problems. Dynamic Systems and Applications, 11, 545-556.

[4] Chan, C.Y. and Vatsala, A.S. (1990) Method of Upper and Lower Solution and Interval Method for Semilinear Euler-Pision-Darboux Equations. Journal of Mathematical Analysis and Applications, 150, 378-381.

http://dx.doi.org/10.1016/0022-247X(90)90110-2

[5] Lakshmikantham, V. and Swansundaran, S. (1987) Interval Method for 1st Order Differential Equations. Applied Mathematics and Computation, 23, 1-5. http://dx.doi.org/10.1016/0096-3003(87)90052-X

[6] Moore, R.E. (1984) Upper and Lower Bounds on Solutions of Differential, Integral Equations without Using Monotonicity, Convexity or Maximum Principle., In: Vichnevetsky, R. and Stepheman, R., Eds., Advances in Computer Methods for Partial Differential Equations, IMACS, 458-461.

[7] Moore, R.E. (1984) A Survey of Interval Methods for Differential Equations. Proceedings of 23rd Conference on Decision and Control, Las Vegas, 12-14 December 1984, 1529-1535.

[8] Walter, W. (1970) Differential and Integral Inequalities. Springer-Verlag, Berlin, Heidelberg.

http://dx.doi.org/10.1007/978-3-642-86405-6

[9] Caprani, O. and Madsen, K. (1975) Contraction Mappings in Interval Analysis. BIT Numerical Mathematics, 15, 362-366. http://dx.doi.org/10.1007/BF01931673

[10] Moore, R.E. (1977) A Test for Existence of Solution to Nonlinear Systems. SIAM Journal on Numerical Analysis, 14, 611-615.

[11] Moore, R.E. (1978) Bounding Sets in Function Spaces with Application to Nonlinear Operator Equations. SIAM Review, 20, 492-512.

[12] Rall, L.B. (1983) Mean-Value and Taylor Forms in Interval Analysis. SIAM Journal on Mathematical Analysis, 14, 223-238.

[13] Rall, L.B. (1982) Integration of Interval Functions II, the Finite Case. SIAM Journal on Mathematical Analysis, 13, 690-698.