A gradient reproducing kernel collocation method for high order differential equations

2019 ◽  
Vol 64 (5) ◽  
pp. 1421-1454 ◽  
Author(s):  
Ashkan Mahdavi ◽  
Sheng-Wei Chi ◽  
Huiqing Zhu
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
High Order

scholarly journals Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1297 ◽  
Author(s):  
Judy P. Yang ◽  
Hon Fung Samuel Lam
Keyword(s):  
Collocation Method ◽  
Reproducing Kernel ◽  
Outer Boundary ◽  
Computational Cost ◽  
Weak Form ◽  
High Order ◽  
Unknown Boundary ◽  
Shape Functions ◽  
High Order Derivative ◽  
Special Cases

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

2017 ◽  
Vol 8 (1-2) ◽  
pp. 40 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Variable Coefficients ◽  
High Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Linear Algebraic Equations

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

Sign in / Sign up

Export Citation Format

Share Document