Page 654 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 654
NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING
Denote by C [0, T ], Rn the space of continuous functions x : [0, T ] → Rn with the norm
x 1 = max x(t) . By a solution to problem (1), (2) we mean a continuously differentiable
t∈[0,T ]
on (0, T ) function x(t) ∈ C [0, T ], Rn that satisfies equation (1) and boundary condition (2).
Employing regular partition ∆N (see [4, 5]) of the interval [0,T] the ∆N general solution
x(∆N , t, λ) to the linear nonhomogenous Fredholm IDE was introduced in cite1. In [7] the
new concept of a general solution to the Fredholm IDE (1) was extended. By substituting the
corresponding expressions of x(∆N , t, λ) into the boundary condition and continuity conditions
of a solution to equation (1) at the interior points of ∆N we construct a system of nonlinear
algebraic equations in parameters. It is proved that the solvability of the BVP is equivalent to
the solvability of this system.
In present communication, an algorithm for finding a numerical solution to BVP (1), (2)
is proposed. To this end, we use the Dzhumabaev parameterization method [3] and results of
[6, 4, 5, 1, 2, 7]. At applying the parameterization method to BVP, the special Cauchy problem
for a system of nonlinear Fredholm IDEs with parameters and a system of nonlinear algebraic
equations in parameters are the intermediate problems. In this case, iterative methods are used
both for solving the special Cauchy problem and for solving the systems of nonlinear algebraic
equations. The algorithm for solving the special Cauchy problem includes two auxiliary prob-
lems: the Cauchy problems for ordinary differential equations and the evaluation of definite
integrals. The accuracy of the method that we propose to solve the BVP (1), (2) depends on
the accuracy of methods applied to the auxiliary problems and does not depend on a number of
the partition subintervals. To solve the Cauchy problems we use the fourth order Runge-Kutta
method and to evaluate definite integrals we use Simpson’s formula. Therefor, the accuracy of
the numerical solution is definite trough the accuracy of these problems.
This research is supported by the Ministry of Education and Science of the Republic Kaza-
khstan Grant AP05132486.
References
[1] D.S. Dzhumabaev, Convergence of iterative methods for unbounded operator equations //
Mat. Zametki, 41: 5 (1987), 356–361.
[2] D.S. Dzhumabaev, On the Conergence of a Modification of the Newton-Kantorovich Method
for Closed Operator Equations // Amer. Math. Soc. Transl., 2 (1989), 95–99.
[3] D.S. Dzhumabaev, Criteria for the unique solvability of a linear boundary-value problem
for an ordinary differential equation // Comput. Math. Math. Phys., 29 (1989), 34–46.
[4] D.S. Dzhumabaev, Necessary and Sufficient Conditions for the Solvability of Linear
Boundary-Value Problems for the Fredholm Integro-differential Equations // Ukrainian
Math. J., 66: 8 (2015), 1200–1219.
[5] D.S. Dzhumabaev, On one approach to solve the linear boundary value problems for Fred-
holm integro-differential equations // J. Comput. Appl. Math., 294 (2016), 342–357.
[6] D.S. Dzhumabaev, New general solutions to linear Fredholm integro-differential equations
and their applications on solving the boundary value problems // J. Comput. Appl. Math.,
327 (2018), 79–108.
[7] D.S. Dzhumabaev, S.T. Mynbayeva, New general solution to a nonlinear Fredholm integro-
differential equation // Eurasian Mathematical Journal., 10: 4 (2019), 24–33.
652
Denote by C [0, T ], Rn the space of continuous functions x : [0, T ] → Rn with the norm
x 1 = max x(t) . By a solution to problem (1), (2) we mean a continuously differentiable
t∈[0,T ]
on (0, T ) function x(t) ∈ C [0, T ], Rn that satisfies equation (1) and boundary condition (2).
Employing regular partition ∆N (see [4, 5]) of the interval [0,T] the ∆N general solution
x(∆N , t, λ) to the linear nonhomogenous Fredholm IDE was introduced in cite1. In [7] the
new concept of a general solution to the Fredholm IDE (1) was extended. By substituting the
corresponding expressions of x(∆N , t, λ) into the boundary condition and continuity conditions
of a solution to equation (1) at the interior points of ∆N we construct a system of nonlinear
algebraic equations in parameters. It is proved that the solvability of the BVP is equivalent to
the solvability of this system.
In present communication, an algorithm for finding a numerical solution to BVP (1), (2)
is proposed. To this end, we use the Dzhumabaev parameterization method [3] and results of
[6, 4, 5, 1, 2, 7]. At applying the parameterization method to BVP, the special Cauchy problem
for a system of nonlinear Fredholm IDEs with parameters and a system of nonlinear algebraic
equations in parameters are the intermediate problems. In this case, iterative methods are used
both for solving the special Cauchy problem and for solving the systems of nonlinear algebraic
equations. The algorithm for solving the special Cauchy problem includes two auxiliary prob-
lems: the Cauchy problems for ordinary differential equations and the evaluation of definite
integrals. The accuracy of the method that we propose to solve the BVP (1), (2) depends on
the accuracy of methods applied to the auxiliary problems and does not depend on a number of
the partition subintervals. To solve the Cauchy problems we use the fourth order Runge-Kutta
method and to evaluate definite integrals we use Simpson’s formula. Therefor, the accuracy of
the numerical solution is definite trough the accuracy of these problems.
This research is supported by the Ministry of Education and Science of the Republic Kaza-
khstan Grant AP05132486.
References
[1] D.S. Dzhumabaev, Convergence of iterative methods for unbounded operator equations //
Mat. Zametki, 41: 5 (1987), 356–361.
[2] D.S. Dzhumabaev, On the Conergence of a Modification of the Newton-Kantorovich Method
for Closed Operator Equations // Amer. Math. Soc. Transl., 2 (1989), 95–99.
[3] D.S. Dzhumabaev, Criteria for the unique solvability of a linear boundary-value problem
for an ordinary differential equation // Comput. Math. Math. Phys., 29 (1989), 34–46.
[4] D.S. Dzhumabaev, Necessary and Sufficient Conditions for the Solvability of Linear
Boundary-Value Problems for the Fredholm Integro-differential Equations // Ukrainian
Math. J., 66: 8 (2015), 1200–1219.
[5] D.S. Dzhumabaev, On one approach to solve the linear boundary value problems for Fred-
holm integro-differential equations // J. Comput. Appl. Math., 294 (2016), 342–357.
[6] D.S. Dzhumabaev, New general solutions to linear Fredholm integro-differential equations
and their applications on solving the boundary value problems // J. Comput. Appl. Math.,
327 (2018), 79–108.
[7] D.S. Dzhumabaev, S.T. Mynbayeva, New general solution to a nonlinear Fredholm integro-
differential equation // Eurasian Mathematical Journal., 10: 4 (2019), 24–33.
652
