A New Treatment of the Decomposition Method for Nonclassical Boundary Value Problems

Yücel, Merve; Mukhtarov, O. Sh.

doi:10.1166/jap.2018.1412

Cited by 4 publications

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“…Yücel and Mukhtarov [16] developed a new generalisation of the DTM for solving nonclassical boundary value problems, which differs from the classical boundary value problems in that, the boundary conditions contain some internal points at which given additional conditions. Al-Saif and Harfash compare the reduced DTM and perturbation-iteration method in [17] solving two-dimensional Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

A New Transformation Method for Solving High-Order Boundary Value Problems

Yücel

Muhtarov

Mukhtarov

2022

Journal of New Theory

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The main purpose of this work is to provide a new approximation method, the so-called parameterised differential transform method (PDTM), for solving high-order boundary value problems (HOBVPs). Our method is based on the classical differential transform method and differs from it by calculating the coefficients of the solution, which has the form of a series. We applied the proposed new method to fourth-order boundary value problems to substantiate it. The resulting solution is graphically compared with the exact solution and the solutions obtained by the classical DTM and ADM methods.

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Section: Introductionmentioning

confidence: 99%

A New Transformation Method for Solving High-Order Boundary Value Problems

Yücel

Muhtarov

Mukhtarov

2022

Journal of New Theory

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“…We want emphasize that two-interval boundary-value problems with additional transmission conditions are much more complicated to solve than one-interval boundary value problems. The existence and uniqueness theorems for the solution of a two-interval boundary value transmission problem (BVTP) can be found in many articles by the first author and his collaborators (see, for example, [19][20][21][22][23][24]. )…”

Section: Introductionmentioning

confidence: 99%

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Mukhtarov

Yücel

2020

Mathematics

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The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

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Discontinuous Sturm-Liouville Problems Involving an Abstract Linear Operator

Mukhtarov¹,

Aydemir²

2020

jaac

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A New Treatment of the Decomposition Method for Nonclassical Boundary Value Problems

Cited by 4 publications

References 0 publications

A New Transformation Method for Solving High-Order Boundary Value Problems

A New Transformation Method for Solving High-Order Boundary Value Problems

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Discontinuous Sturm-Liouville Problems Involving an Abstract Linear Operator

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