Exact analytic method for solving variable coefficient differential equation

1989 ◽  
Vol 10 (10) ◽  
pp. 885-896 ◽  
Author(s):  
Ji Zhen-yi ◽  
Yeh Kai-yuan
Keyword(s):  
Differential Equation ◽  
Variable Coefficient ◽  
Analytic Method ◽  
Exact Analytic Method

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.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

2015 ◽  
Vol 70 (6) ◽  
pp. 445-450 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Variable Coefficient ◽  
Nonlinear Partial Differential Equation ◽  
Direct Methods ◽  
Periodic Wave ◽  
Variable Coefficients ◽  
Nonlinear Phenomena ◽  
Similarity Reduction ◽  
Petviashvili Equation

AbstractIn this paper, the generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (VCKPE), which can describe nonlinear phenomena in fluids or plasmas, is studied by using two different Clarkson and Kruskal (CK) direct methods, namely, the classical CK and the modified enlarged CK method. A similarity reduction to a (2+1)-dimensional nonlinear partial differential equation and a direct similarity reduction to a nonlinear ordinary differential equation are obtained, respectively. By solving the reduced ordinary differential equation, new solitary, periodic, and singular solutions for the VCKPE are obtained. Some figures for the soliton and periodic wave solutions are given to reflect the effect of the variable coefficients on the solution propagation. Finally, the comparison between the two different CK techniques indicates that the modified enlarged CK technique is clearly more powerful and simple than the classical CK technique.

scholarly journals Parametric Excitation and Evolutionary Dynamics

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Rocio E. Ruelas ◽  
David G. Rand ◽  
Richard H. Rand
Keyword(s):  
Differential Equation ◽  
Parametric Excitation ◽  
Variable Coefficient ◽  
Evolutionary Dynamics ◽  
Replicator Equation ◽  
Periodic Coefficients ◽  
Forcing Function ◽  
Governing Differential Equation ◽  
Social Applications ◽  
Dynamics Problems

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

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.

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