Artificial Neural Network Technique for Solution of Nonlinear Elliptic Boundary Value Problems

Yadav, Neha; Yadav, Anupam; Deep, Kusum

doi:10.1007/978-81-322-2217-0_10

Cited by 6 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, which means the pseudoconcavity of the operator T . Further, from the inequalities (11), (12) it follows that for all…”

Section: Methodsmentioning

confidence: 99%

“…where the function ˆ( , , ) f v w x , continuous in the set of variables x , v , w , monotonically increases with respect to v and monotonically decreases with respect to w for all ∈ Ω x ; d) if inequality (11) holds, then it is a pseudo-concave and even 0 u -pseudo-concave operator, where the function 0 ( ) u x has the form (6). Further one will assume that the operator T of the form ( 5) is heterotone one with the companion operator of the form (8).…”

Section: So For All ∈ ωmentioning

confidence: 99%

“…and condition (11) of 0 u -pseudo-concavity takes the following form: for any positive number u and for any (0, 1) τ∈…”

Section: So For All ∈ ωmentioning

confidence: 99%

“…The pseudo-concavity condition (11), written for the function ( , ) f u x of the form (38), leads to the inequality 1 1…”

Section: For a Function ( )mentioning

confidence: 99%

“…Currently there are many methods for the numerical analysis of the boundary value problems for semilinear elliptic equations. Among them, one can single out, in particular, the methods of finite differences, finite elements, boundary integral equations, artificial neural network technique [1,[5][6][7][8][9][10][11] or successive approximations with two-sided convergence [12][13][14]. The methods of the last group allow to construct two sequences of functions that approximate the desired solution of the problem from below and from above, respectively.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Two-Sided Approximations Method Based on the Green’s Functions Use for Construction of a Positive Solution of the Dirichle Problem for a Semilinear Elliptic Equation

Gybkina

Lamtyugova

Sidorov

2021

RIC

View full text Add to dashboard Cite

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

show abstract

“…, which means the pseudoconcavity of the operator T . Further, from the inequalities (11), (12) it follows that for all…”

Section: Methodsmentioning

confidence: 99%

Section: So For All ∈ ωmentioning

confidence: 99%

“…and condition (11) of 0 u -pseudo-concavity takes the following form: for any positive number u and for any (0, 1) τ∈…”

Section: So For All ∈ ωmentioning

confidence: 99%

“…The pseudo-concavity condition (11), written for the function ( , ) f u x of the form (38), leads to the inequality 1 1…”

Section: For a Function ( )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two-Sided Approximations Method Based on the Green’s Functions Use for Construction of a Positive Solution of the Dirichle Problem for a Semilinear Elliptic Equation

Gybkina

Lamtyugova

Sidorov

2021

RIC

View full text Add to dashboard Cite

show abstract

Medical Diagnosing of Canine Diseases Using Genetic Programming and Neural Networks

Mahapatra¹

2017

Nature Inspired Computing

View full text Add to dashboard Cite

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

Ahmad

Raja

Ramos

et al. 2020

Neural Comput & Applic

View full text Add to dashboard Cite

In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear twopoint boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA-AST, PS-AST and SA-AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance.

show abstract

Artificial Neural Network Technique for Solution of Nonlinear Elliptic Boundary Value Problems

Cited by 6 publications

References 12 publications

Two-Sided Approximations Method Based on the Green’s Functions Use for Construction of a Positive Solution of the Dirichle Problem for a Semilinear Elliptic Equation

Two-Sided Approximations Method Based on the Green’s Functions Use for Construction of a Positive Solution of the Dirichle Problem for a Semilinear Elliptic Equation

Medical Diagnosing of Canine Diseases Using Genetic Programming and Neural Networks

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

Contact Info

Product

Resources

About