1,210 research outputs found
Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations
In this paper, we use a numerical method that involves hybrid and block-pulse
functions to approximate solutions of systems of a class of Fredholm and
Volterra integro-differential equations. The key point is to derive a new
approximation for the derivatives of the solutions and then reduce the
integro-differential equation to a system of algebraic equations that can be
solved using classical methods. Some numerical examples are dedicated for
showing efficiency and validity of the method that we introduce
Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations
Due to the ability of function representation, hybrid functions and wavelets have a
special position in research. In this thesis, we state elementary definitions, then we
introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby-
shev, sine-cosine and linear Legendre multi wavelets. The construction of most
wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them.
The properties of the Floor function are used to and a function which is one on the
interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters
lead us to get similar function corresponding to the interval [a; b). These functions
and their combinations enable us to represent the stepwise functions as a function of
floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet,
Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new
representations of these functions.
The main advantage of the wavelet technique for solving a problem is its ability
to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential
and Fredholm integral equations of the second kind. We also use collocation points
and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic
fields. Illustrative examples are included to reveal the sufficiency of the technique.
In linear integro-differential equations and Fredholm integral equations of the second
kind cases, comparisons are done with CAS wavelets and differential transformation
methods and it shows that the accuracy of these results are higher than them.
Homotopy Analysis Method (HAM) is an analytic technique to solve the linear
and nonlinear equations which can be used to obtain the numerical solution too.
We extend the application of homotopy analysis method for solving Linear integro-
differential equations and Fredholm and Volterra integral equations. We provide
some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW,
LLMW and CAS wavelets methods. For future studies, some problems are proposed
at the end of this thesis
A hybrid functions method for solving linear and non-linear systems of ordinary differential equations
In the present paper, we use a hybrid method to solve linear or non-linear systems of ordinary differential equations (ODEs). By using this method, these systems are reduced to a linear or non-linear system of algebraic equations. In error discussion of the suggested method, an upper bound of the error is obtained. Also, to survey the accuracy and the efficiency of the present method, some examples are solved and comparisons between the obtained results with those of several other methods are carried out
Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function
تقدم هذه الورقة البحثية طريقة لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي الثلاثي الأبعاد غير الخطي في R3. حيث يتم استخدام مفهوم (Majorant function) وباستخدام طريقة نيوتن المعدلة لتحويل مؤثر فولتيرا التكاملي الثلاثي الأبعاد غير الخطي إلى متتالية لمؤثر فولتيرا التكاملي الثلاثي الأبعاد الخطي ومن يتم استخدام طريقة (Gaussian-Legendre) التربيعية لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي الثلاثي الأبعاد الخطي من خلال التعامل مع نظام جبري خطي.تم مناقشة وجود ووحدانية الحل للطريقة المستخدمة مع اعطاء أمثلة توضيحية لإظهار دقة وكفاءة الطريقة.In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.
Mathematical Subject Classification (2010): 45P05, 45G10, 47H9
Numerical Approximation of Nonlinear Stochastic Volterra Integral Equation using Walsh Function
This article proposes an efficient numerical method for solving nonlinear
stochastic Volterra integral equations using the operational matrices of the
Walsh function and the collocation method. In this method, a nonlinear
stochastic Volterra integral equation is reduced to a system of algebraic
equations, which are then solved to obtain an approximation of the solution.
Error analysis has been performed, confirming the effectiveness of the
methodology, which results in a linear order of convergence. Examples were
computed to demonstrate the efficacy and precision of the method.Comment: arXiv admin note: substantial text overlap with arXiv:2305.16678,
arXiv:2305.0082
Computational Block-Pulse Functions Method for Solving Volterra Integral Equations with Delay
يتمثل الهدف من هذا العمل في اتباع نهج طريقة الضغط النبضي في الحل العددي لمعادلات فولتيرا التكاملية مع التأخير. تستخدم هذه الطريقة للحصول على حل رقمي. علاوة على ذلك، تتم كتابة البرنامج بلغة MATLAB. تم عمل تحليل للخطأ وتم توضيح التطبيقات من خلال المثال التوضيحيThe aim of this work is to present method of the Block-pulse function approach to numerical solution of Volterra integral equations with delay. This method is used to obtain numerical solution. Moreover, programs for his method is written in MATLAB language. An error analysis is worked out and applications demonstrated through illustrative example
Legendre Wavelets Method for Solving Fractional Population Growth Model in a Closed System
A new operational matrix of fractional order integration for Legendre wavelets is derived. Block pulse functions and collocation method are employed to derive a general procedure for forming this matrix. Moreover, a computational method based on wavelet expansion together with this operational matrix is proposed to obtain approximate solution of the fractional population growth model of a species within a closed system. The main characteristic of the new approach is to convert the problem under study to a nonlinear algebraic equation
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