IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT

2016 ◽  
Vol 78 (6-5) ◽  
Author(s):  
Nurul Akmal Mohamed ◽  
Nur Fadhilah Ibrahim ◽  
Mohd Rozni Md Yusof ◽  
Nurul Farihan Mohamed ◽  
Nurul Huda Mohamed
Keyword(s):  
Differential Equation ◽  
Quadrature Formula ◽  
Spectral Properties ◽  
Variable Coefficient ◽  
Logarithmic Singularity ◽  
Neumann Series ◽  
Numerical Results ◽  
Analytic Method ◽  
Elliptic Type ◽  
Integro Differential Equation

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

scholarly journals A Convolution Type Nonlinear Integro-Differential Equation with a Variable Coefficient and an Inhomogeneity in the Linear Part

2020 ◽  
pp. 16-27
Author(s):  
С.Н. Асхабов
Keyword(s):  
Differential Equation ◽  
Variable Coefficient ◽  
Linear Part ◽  
Convolution Type ◽  
Integro Differential Equation

Изучается вольтерровское интегро-дифференциальное уравнение типа свертки со степенной нелинейностью, переменным коэффициентом $a(x)$ и неоднородностью $f(x)$ в линейной части, которое тесно связано с соответствующим нелинейным интегральным уравнением, возникающим при исследовании инфильтрации жидкости из цилиндрического резервуара в изотропную однородную пористую среду, при описании процесса распространения ударных волн в трубах, наполненных газом, при решении задачи о нагревании полубесконечного тела при нелинейном теплопередаточном процессе, в моделях популяционной генетики и других. Важно отметить, что в связи с указанными и другими приложениями особый интерес представляют непрерывные положительные при $x>0$ решения интегрального уравнения. На основе полученных точных нижней и верхней априорных оценок решения интегрального уравнения мы строим весовое полное метрическое пространство $P_b$, инвариантное относительно нелинейного интегрального оператора свертки, порожденного этим уравнением, и, применяя метод весовых метрик (аналог метода Белицкого), доказываем глобальную теорему о существовании и единственности решения изучаемого нелинейного интегро-дифференциального уравнения как в пространстве $P_b$, так и во всем классе $Q_0^1$ непрерывно дифференцируемых положительных при $x>0$ функций. Показано, что решение может быть найдено в пространстве $P_b$ методом последовательных приближений пикаровского типа. Для последовательных приближений получены оценки скорости их сходимости к точному решению в терминах весовой метрики пространства~$P_b$. В частности, при $f(x)=0$ из этой теоремы вытекает, что соответствующее однородное нелинейное интегро-дифференциальное уравнение, в отличие от линейного случая, имеет нетривиальное решение. Приведены также примеры, иллюстрирующие полученные результаты.

Nonlinear convolution integro-differential equation with variable coefficient

2021 ◽  
Vol 24 (3) ◽  
pp. 848-864
Author(s):  
Sultan N. Askhabov
Keyword(s):  
Differential Equation ◽  
Variable Coefficient ◽  
A Priori Estimates ◽  
Complete Metric Space ◽  
Continuous Functions ◽  
Heat Transfer Process ◽  
Fractional Integrals ◽  
Successive Approximation Method ◽  
Infinite Body ◽  
Integro Differential Equation

Abstract For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C[0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.

The Elastohydrodynamic Lubrication of a Thin Layer

1973 ◽  
Vol 95 (3) ◽  
pp. 381-385 ◽  
Author(s):  
H. D. Conway ◽  
P. A. Engel
Keyword(s):  
Differential Equation ◽  
Film Thickness ◽  
Thin Layer ◽  
Elastic Layer ◽  
Variable Viscosity ◽  
Elastohydrodynamic Lubrication ◽  
Numerical Results ◽  
Integral Constraint ◽  
Integro Differential Equation ◽  
Thin Elastic Layer

The elastohydrodynamic lubrication problem of a thin elastic layer pressed between two cylinders which rotate in opposite directions is discussed. The analysis leads to an integro-differential equation for the film thickness, subject to an integral constraint. Numerical results are given for both the isoviscous and variable viscosity cases.

scholarly journals An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

2020 ◽  
pp. 86-96
Author(s):  
Galina A. Rasolko ◽  
Sergei M. Sheshko
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Electromagnetic Waves ◽  
Logarithmic Singularity ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Solution Function ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed.  This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative.  The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals INVESTIGATION OF SOME CLASSES OF SECOND ORDER PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH A POWERLOGARITHMIC SINGULARITY IN THE KERNEL

2020 ◽  
pp. 40-54
Author(s):  
Sarvar K. ZARIFZODA ◽  
◽  
Raim N. ODINAEV ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Operators ◽  
Logarithmic Singularity ◽  
Second Order ◽  
Fredholm Type ◽  
Power Singularity ◽  
Characteristic Equations ◽  
Integro Differential Equation ◽  
The Given

For a class of second-order partial integro-differential equations with a power singularity and logarithmic singularity in the kernel, integral representations of the solution manifold in terms of arbitrary constants are obtained in the class of functions vanishing with a certain asymptotic behavior. Although the kernel of the given equation is not a Fredholm type kernel, the solution of the studied equation in a class of vanishing functions is found in an explicit form. We represent a second-order integro-differential equation as a product of two first-order integro-differential operators. For these one-dimensional integro-differential operators, in the cases when the roots of the corresponding characteristic equations are real and different, real and equal and complex and conjugate, the inverse operators are found. It is found that the presence of power singularity and logarithmic singularity in the kernel affects the number of arbitrary constants in the general solution. This number, depending on the roots of the corresponding characteristic equations, can reach nine. Also, the cases when the given integro-differential equation has a unique solution are found. The correctness of the obtained results with the help of the detailed solutions of concrete examples are shown. The method of solving the given problem can be used for solving model and nonmodel integro-differential equations with a higher order power singularity and logarithmic singularity in the kernel.

AN INTEGRO-DIFFERENTIAL EQUATION ARISING IN RADIONUCLIDE TRANSPORT THROUGH FRACTURED ROCK

1993 ◽  
Vol 03 (05) ◽  
pp. 641-654 ◽  
Author(s):  
J.S. CASSELL ◽  
M.M.R. WILLIAMS
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Physical Interpretation ◽  
Special Form ◽  
Fractured Rock ◽  
Numerical Results ◽  
Radionuclide Transport ◽  
Integro Differential Equation ◽  
Redistribution Function

An integro-differential equation arising in the transport of radionuclides through fractured rock has been studied. Using a special form for the velocity redistribution function at fracture intersections, it has been possible to obtain an analytical solution. Numerical results are given to illustrate the theory and its physical interpretation.

scholarly journals An hp-adaptive strategy for the solution Of the exact kernel curved wire pocklington Equation

2008 ◽  
Vol 8 (1) ◽  
pp. 39-59
Author(s):  
P.W. HEMKER ◽  
D.J.P. LAHAYE
Keyword(s):  
Differential Equation ◽  
Adaptive Strategy ◽  
Adaptive Method ◽  
Numerical Results ◽  
Wire Antenna ◽  
Thin Wire ◽  
Discrete Solution ◽  
Hierarchical Basis ◽  
Integro Differential Equation ◽  
Adaptive Technique

AbstractIn this paper we introduce an adaptive method for the numerical solu-tion of the Pocklington integro-differential equation with exact kernel for the current induced in a smoothly curved thin wire antenna. The hp-adaptive technique is based on the representation of the discrete solution, which is expanded in a piecewise p-hierarchical basis. The key element in the strategy is an element-by-element criterion that controls the h- or p-refinement. Numerical results demonstrate both the simplicity and efficiency of the approach

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