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Application of HAM for Nonlinear Integro-Differential Equations of Order Two

Journal of applied mathematics and Physics 2023年第1期11卷 55-68页

作者： Zainidin Eshkuvatov Davron Khayrullaev Muzaffar Nurillaev Shalela Mohd Mahali Anvar Narzullaev Faculty of Ocean Engineering Technology and Informatics University Malaysia Terengganu (UMT) Terengganu Malaysia Faculty of Applied Mathematics and Intellectual Technologies National University of Uzbekistan Tashkent Uzbekistan Faculty of Physics and Mathematics Tashkent State Pedagogical University (TSPU) Tashkent Uzbekistan Faculty of Science and Technology University Sains Islam Malaysia (USIM) Bandar Baru Nilai Malaysia

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

关键词： Integral-Differential Equations Homotopy Analyses Method Iterative System Algebraic Equations Gauss-Legendre Quadrature Formula

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Stability of the difference scheme for the shallow water equation

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AIP Conference Proceedings 2023年第1期2781卷

作者： Mirzoali Khudoyberganov Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan NUUz Tashkent Uzbekistan

The shallow water equation is converted to a divergent form using special substitutions. A priori estimate is obtained for the solution of the differential problem by constructing an energy integral for the resulting equation. A differential scheme is also constructed that approximates the differential problem under consideration, and the stability of this differential scheme is proved.

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The automatic quadrature scheme for the first order weighted singular integrals

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AIP Conference Proceedings 2023年第1期2746卷

作者： Zainidin K. Eshkuvatov Sayfiddin Bakhramov Faculty of Ocean Engineering Technology and Informatics Universiti Malaysia Terengganu (UMT) Terengganu Malaysia Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan (NUUz) Tashkent Uzbekistan

In this note, bounded and unbounded weighted Cauchy type singular integrals (SIs) with a variable limit of integration are considered. Automatic quadrature schemes (AQS) together with truncated series of first and second kind Chebyshev polynomials with corresponding weighted functions are applied to get high order approximate solution. Some examples are provided to validate the proposed method. It is shown that when the density function is a polynomial function the AQS gives an exact solution, and for rational function, the error term of AQS decreases gradually when the number of nodes increases.

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Error estimates for one local interpolation cubic spline with the maximum approximation order of O(h3) for W1[a, b] classes

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AIP Conference Proceedings 2023年第1期2484卷

作者： Sayfiddin A. Bakhromov Zainidin K. Eshkuvatov 1Faculty of Applied Mathematics and Intellectual Technologies National University of Uzbekistan (NUUz) Tashkent Uzbekistan. 2Faculty of Ocean Engineering Technology and Informatics University Malaysia Terengganu (UMT) 21300 Kuala Terengganu Malaysia.

It is known that local splines have been widely used for processing of various signals such as restoration of geophysical, biomedical signals. Moreover, it uses for the approximate calculation of regular and singular integrals, integrals of the Fourier types. Using such splines, we can build effective quadrature and cubature formulas. It uses for approximate solution of regular and singular integrals equations. In this note, error estimation of local spline functions is obtained in the class of function W1[a, b]. Moreover, comparisons with other splines are shown in graphical experiments. On the other hand, it is shown that the order of approximation of local spline function is higher than the Ryabenky and Grebennikov cubic splines. It is shown that local spline function in all cases achieved tangible results in comparison with the classical polynomial approximation. It is well known that the maximum order of approximation by the Ryabenky and Grebennikov cubic splines is O(h2) and maximum order of approximation of the local cubic spline function is O(h3), and the interpolation condition is satisfied. Numerical experiments are also in the line of theoretical findings.

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New development of HAM for approximating linear integro-differential equation of order two

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AIP Conference Proceedings 2023年第1期2484卷

作者： Z. K. Eshkuvatov M. E. Nurillaev B. S. Abdullaeva D. B. Khayrullaev Sh. M. Mahalis 1Faculty of Ocean Engineering Technology and Informatics University Malaysia Terengganu (UMT) Kuala Nerus Terengganu Malaysia. 2Independent researcher Faculty of Applied Mathematics and Intellectual Technologies National University of Uzbekistan (NUUz) Tashkent Uzbekistan. 3Faculty of Physics and Mathematics Tashkent State Pedagogical University named after Nizami (TSPU) Tashkent Uzbekistan.

In many problems of mathematical physics, theory of elasticity, visco-dynamics fluid and mixed problems of mechanics of continuous media reduces to the integral and intro-differential equation (Volterra or Fredholm type) first or second kind. In the early XXth (around 1910) century, to solve some problems in the field of mechanics Volterra had to solve integro-differential equations (IDEs) with variable boundaries. One of the popular methods for solving Volterra IDEs is the method of quadratures when problem of consideration is a linear problem. When IDEs are nonlinear or integral kernel are complicated (singular or nonlinear) then quadrature rule is not most suitable therefore other type of methods are needed to develop. In this note, we have used the standard and new-development of homotopy analysis method (ND-HAM) for the general second order linear integral-differential equations (IDEs) of the second kind. Both methods reduce IDEs into iterative sequence of algebraic integral equations with known integrant function. Gauss-Legendre quadrature formulas are applied for kernel integrations. Obtained system of algebraic equations are solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and reliable.

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Numerical approach for nonlinear system of Fredholm-Volterra integral equations

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AIP Conference Proceedings 2021年第1期2365卷

作者： Zainidin Eshkuvatov Husnida Mamatova Shahrina Ismail Ilyani Abdullah Rakhmatillo Aloev 1)Faculty of Ocean Engineering Technology and Informatics (FOETI) University Malaysia Terengganu (UMT) Kuala Terengganu Malaysia 2)Independent Researcher Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan (NUUz) Tashkent Uzbekistan 3)Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan (NUUz) Tashkent Uzbekistan 4)Faculty of Science and Technology University Sains Islam Malaysia (USIM) Negeri Sembilan Malaysia

In this note, the homotopy analysis method (HAM) is applied as a tool for solving the system of non-linear Fredholm-Volterra integral equations. The generalized chain rule is implemented for differentiation of the non-linear kernel functions with many variables, and the non-linear problem is reduced into a sequence of known non-linear integral equa- tions. The proposed method is compared with the homotopy perturbation method (HPM) given in Yusufogli [20] and Biazar and Ghazyini [21]. Numerical results reveal that the proposed method is very effective with high accuracy and dominated the HPM.

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Numerical solution of two-dimensional coupled thermoplastic problems

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AIP Conference Proceedings 2022年第1期2637卷

作者： A. A. Khaldjigitov Yu. S. Yusupov U. E. Adambaev D. A. Sagdullaeva 1Faculty of Mathematics National University of Uzbekistan Tashkent 100174 Uzbekistan 2Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan Tashkent 100174 Uzbekistan 3Institute of Mechanics and Seismic Stability of Structure named after M.T. Urazbaev Academy of Sciences of the Republic of Uzbekistan 100125 Tashkent Uzbekistan

Coupled dynamic thermoplastic boundary value problems in two dimensions based on strain space and deformation theories of plasticity are formulated. The explicit and implicit types of finite difference equations are constructed. Numerical solution of the explicit finite difference equations is obtained by the application of recurrent formulas, whereas the implicit scheme is obtained by utilizing the elimination method. Comparison of the numerical results obtained by the explicit and implicit schemes show a good coincidence.

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Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order

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AIP Conference Proceedings 2021年第1期2365卷

作者： Zainidin Eshkuvatov Zaid Laadjal Shahrina Ismail 1)Faculty of Ocean Engineering Technology and Informatics (FOETI) Universiti Malaysia Terengganu (UMT) Kuala Terengganu Malaysia 2)Independent Researcher Faculty of Applied Mathematics and Intellectual Technology National University of Uzbekistan (NUUz) Tashkent Uzbekistan 3)Department of Mathematics and Computer Sciences ICOSI Laboratory Abbes Laghrour University Khenchela 40004 Algeria 4)Faculty of Science and Technology Universiti Sains Islam Malaysia (USIM) Negeri Sembilan Malaysia

In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering. Many essential phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are well described by differential equations of fractional order (Podlubny [8]). In certain circumstances, many exact solutions for linear fractional differential equation had been found. However, in general, there exists no method that yields an exact solution for nonlinear fractional differential equations. In this note, we consider the following nonlinear mixed Volterra-Fredholm integro-differential equation (VFIDEs) of fractional order. cD0+αu(t)φ(t)+λ∫0t ∫0Tk(x,s)F(u(s))dxds,(1)u(i)(0)=bi, i=0,…,n−1, n−1<α≤n,(2)where t ∈ Ω = [0; T], k : Ω × Ω → ℝ, ϕ : Ω → ℝ are known functions, F : C(Ω, ℝ) → ℝ is nonlinear function, and u(s) is unknown function to be determined, bi (i = 0,…, n − 1) and λ are constants, cD0+α is the Caputo fractional derivative of order α. At first, Eq. (1)-(2) is written into operator form and applied standard homotopy analysis method (Liao, 1992) to reduce it into a sequence of known integral equation problems. Solving the latter equations step by step using the truncated Taylor series, an approximate solution is obtained. Description of the method and discretization of the problems together with the control convergence parameter are thoroughly studied. In addition, one example is illustrated to show the accuracy and validity of the proposed approach. Numerical results reveal that the proposed method is highly accurate and quickly approaches the exact solution when the number of iteration increases.

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