#### Title

Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations

#### Date of Award

Summer 8-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Committee Chair

Dr. Ching-Shyang Chen

#### Committee Chair Department

Mathematics

#### Committee Member 2

Dr. Haiyan Tian

#### Committee Member 2 Department

Mathematics

#### Committee Member 3

Dr. Huiqing Zhu

#### Committee Member 3 Department

Mathematics

#### Committee Member 4

Dr. Zhaoxian Zhou

#### Committee Member 4 Department

Computing

#### Abstract

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate.

#### ORCID ID

orcid.org/0000-0001-5529-2811

#### Copyright

Balaram Khatri Ghimire, 2016

#### Recommended Citation

Khatri Ghimire, Balaram, "Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations" (2016). *Dissertations*. 382.

http://aquila.usm.edu/dissertations/382