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An analytical solution technique that is fast and efficient and provides accurate results is developed for the fourth-order non-linear differential equation describing the transverse vibrations of a beam on a two parameter elastic... more
This paper is devoted to investigate the relationship in the performance of the four kinds of shifted Chebyshev polynomial for the comparison purpose, in fractional linear differential equation with constant coefficient involving the... more
The Galerkin method is a numerical technique used to approximate solutions to Partial Differential Equations (PDEs) or integral equations. To this end, this study employed the Galerkin method to address Multi-Order Fractional Differential... more
Integro-differential equations are pivotal in modeling various phenomena in physics and engineering, where the system's current state depends on its history.This study explores the approximation of second-order mixed Volterra-Fredholm... more
The relatively efficient and accurate Adomian modified decomposition method (AMDM) is used in this paper to investigate the free vibrations of Euler-Bernoulli beams, with a single section discontinuity present, and resting on a... more
The initialization of equation-based differential-algebraic system models, and more in general the solution of many engineering and scientific problems, require the solution of systems of nonlinear equations. Newton-Raphson's method is... more
In many engineering applications, such as satellite dynamics or various case-studies of the specific locomotion patterns in mechatronics and biomechanics, motion integrals of the system need to be conserved during numerical integration in... more
In this study the application of a newly developed efficient method namely, optimal homotopy analysis method (OHAM) has been illustrated for solving nonlinear singular boundary value problems (SBVPs) which frequently arise in chemical and... more
In this study the application of a newly developed efficient method namely, optimal homotopy analysis method (OHAM) has been illustrated for solving nonlinear singular boundary value problems (SBVPs) which frequently arise in chemical and... more
This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main... more
In this paper, we devise a layer stripping algorithm for any dynamical isotropic elasticity system of equations in three-dimensional space. We give an explicit reconstruction of both Lamé moduli and the density, as well as their... more
The paper develops and implements a numerical method for solving fractional order Fredholm Volterra integro differential equations with Dirichlet boundary conditions using the shifted Legendre collocation method. The proposed method is... more
In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (V-FIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the... more
This article introduces a numerical method that uses hybrid functions for approximating solutions of systems of Fredholm integro-differential equations of the second kind. This method reduces a system of Fredholm integro-differential... more
In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects,... more
Laminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous–inviscid interaction concept. The ‘triple-deck model’ is used to... more
The diffraction of high frequency cylindrical electromagnetic waves by step discontinuities is investigated rigorously by using the Fourier transform technique in conjunction with the mode matching method. The hybrid method of formulation... more
W pracy przedstawiono metodę modelowania numerycznego procesu zapł onu pojedynczej, kulistej czą stk i paliwa stał ego. Wychodząc z cieplno-dyfuzyjnej teorii zapł onu, zagadnienie rozwią zan o metodą róż ni c skoń czonych , stosując... more
In this article, we consider Cauchy problem for the nonlinear parabolic-hyperbolic partial differential equations. These equations are solved by He-Laplace method.. It is shown that, in He-Laplace method, the nonlinear terms of... more
In this paper, we report new three level implicit super stable methods of order two in time and four in space for the solution of hyperbolic damped wave equations in one, two and three space dimensions subject to given appropriate initial... more
A one dimensional analytical model is developed for the steady state, axisymmetric flow of damp powder within a rotating impervious cone. The powder spins with the cone but migrates up the wall of the cone (along a generator) under... more
A new approach to find the optimal solution of linear time-invariant scaled systems using the Chebyshev wavelets is proposed. The operational matrix of stretch is derived and together with the operational matrix of integration are used to... more
We use Chebyshev wavelets on the interval {Ŵ ŵ{ to solve the nonlinear variational problems with moving boundary conditions.An operational matrix of integration is introduced and utilized to reduce the variational problems to the solution... more
Optimal control of linear time-varying systems using the Chebyshev wavelets (a comparative approach)
This paper extends the application of continuous Chebyshev wavelet expansions to find the optimal solution of linear timevarying systems using two different approaches. By using the product of two time functions together with the... more
A numerical technique based on Legendre Polynomials for finding the optimal control of nonlinear systems with quadratic performance index is presented. An operational matrix of integration and product matrix are introduced and are used to... more
A new approach to find the approximate solution and the optimal control of linear time-invariant scaled systems using the Legendre wavelets is proposed. The operational matrix of stretch is derived and together with the operational matrix... more
We use Chebyshev wavelets on the interval ����� to solve the nonlinear variational problems with moving boundary conditions.An operational matrix of integration is introduced and utilized to reduce the variational problems to the solution... more
Numerical methods are used to approximate solutions of equations when exact solutions can not be determined via algebraic methods. They construct successive approximations that converge to the exact solution of an equation or system of... more
In this paper a three-step two hybrid block method with two offgrid hybrid points chosen within interval [Xn,Xn+1] and [Xn+1,Xn+2] was developed to solve second Order Ordinary Differential Equations directly, using the power series as the... more
Spectral methods solve elliptic PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite... more
In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on... more
We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two-or three-dimensional domain defined... more
This work develops an approximate technique based on the hybrid of block-pulse functions and Chelyshkov polynomials for solving a class of linear optimal control problems (OCPs). In the initial step of the method, the state and control... more
Causal Topology: Designing Recursive Civilization presents a groundbreaking theoretical framework for understanding the interplay between geometric structures, causality, and complex systems. Drawing upon concepts from thermodynamics,... more
Implicit integration schemes for large systems of nonlinear ODEs require, at each integration step, the solution of a large nonlinear system. Typically, the nonlinear systems are solved by an inexact Newton method that leads to a set of... more
In the current paper, we present an efficient direct scheme for weakly singularVolterra integro-differential equations arising in theory of anomalous diffusion. The behav-ior of the system demonstrating the anomalous diffusion is... more
In this paper, we consider an inverse boundary value problem for two-dimensional heat equation in an annular domain. This problem consists of determining the temperature on the interior boundary curve from the Cauchy data (boundary... more
Numerical solution of differential-algebraic equations with Hessenberg index-3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.
Nonlinear chaotic finance systems are represented by nonlinear ordinary differential equations and play a significant role in micro-and macroeconomics. In general, these systems do not have exact solutions. As a result, one has to resort... more
The relatively efficient and accurate Adomian modified decomposition method (AMDM) is used in this paper to investigate the free vibrations of Euler-Bernoulli beams, with a single section discontinuity present, and resting on a... more
𝐺 ̇(𝑡) + 𝐺(𝑡 -𝜏)𝐷 + 𝐷 𝑇 𝐺(𝑡) -𝐺(𝑡)𝑃𝐺(𝑡) + 𝑆(𝑡) = 0, 𝑡 ∈ [𝑐, 𝑇], 1 where 𝜏 > 0 constant, 𝑐 ∈ 𝑅 . and 𝐺, 𝑃, 𝑆 and 𝐷 are 𝑚 × 𝑚 matrices; 𝑃 = 𝑃 𝑇 , 𝑆 = 𝑆 𝑇 , 𝜏 = 1, and the initial matrix function 𝐺(𝑡) = 𝑍 0 (𝑡), 𝑐 -𝜏 ≤ 𝑡 ≤ 𝑐, 𝑐 a positive .... more
In this paper we represent a new form of condition for the consistency of the matrix equation AXB = C. If the matrix equation AXB = C is consistent, we determine a form of general solution which contains both reproductive and... more
Free vibration analysis of composite cylindrical shells with different boundary conditions is presented in this paper using differential quadrature method (DQM). Equations of motion are derived based on first order shear deformation... more
Despite numerous studies in the field of creating rubber products, to date, the choice of material for the manufacture of packer sealing elements has not found its solution. This issue becomes especially important drilling deep and ultra... more
A method for the solution of systems of linear algebraic equations is presented. It is derived using an approach based on linear estimation theory and is directly related to a generalization of Huang's method (1975) proposed by .... more
This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe.... more
This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the... more
12. Kokotova, Ye.V. Approximation of a bounded solution of systems of linear ordinary differential equations with unbounded coefficients, Mathematical Journal, 5: 3(17) (2005), 40-43 (in Russian). Кокотова Е.В. Сингулярлы дифференциалдық... more