Problems: 6.1, 6.5, 6.6 I am stuck on these and need help to figure them out tha

Question

Problems: 6.1, 6.5, 6.6

I am stuck on these and need help to figure them out

thank you

61 What is the grid spacing Ar if you discretize the space-re [-1, 1 ] with 51 nodes? 6.2 What is the equation for ci for the solution of the diffusion equation d2cdr2-0 with a no-flux boundary condition at the left boundary? 6.3 In a one-dimensional boundary value problem with a no-flux condition on the left boundary, what is the relationship between the fictitious node and the nodes in the domain? 64 Consider a coupled system of boundary value problems for x and y. If the unknowns are in a vector z in an order that preserves the band structure, what are the first five entries in z? 6.5 Consider the differential equation d'y dx2 subject to(0) = 0 and y(1) = 1 If this equation is solved using centered finite differences and three nodes, what is the value of interor node, y2? 6.6 Use Taylor series expansions to derive the centered finite difference formula for the third derivative, = de

Explanation / Answer

Numerical solutions to (1) ,( 2) involve discretizations. Agarwal’s paper [l] provides some excellent examples illustrating that when a continuous boundary value problem is discretized, the nature of the solution may change. For example, extra solutions (or “ghost” solutions) to (3),(4) can appear which may be “large” and “irrelevant” to the continuous problem as the step size tends to zero (see [2]). M oreover, [l] gives an example where the continuous problem possesses a solution, whereas the discrete problem does not. It is therefore natural to ask: under which conditions does the resultant difference equation actually have a solution? Gaines [2] pioneered research regarding the above question for the scalar case, while Henderson and Thompson [3] extended the existence results of Gaines’ to the case of fully nonlinear boundary conditions. Thompson and Tisdell [4] broadened the above results to apply to fully coupled systems of equations based on discrete counterparts to some of the results in [5]. This paper studies the aforeposed question for the discretization (3),(4) by presenting natural discrete analogues to some of the revolutionary a priori bound conditions and existence results due to Hartman [6] for systems of second-order ordinary differential equations. Our work is organised as follows. A condition from [6], which assures a prior-i bounds on derivatives of solutions to (l), is introduced in Section 3 and successfully applied to the discrete problem. The resulting discrete analogue guarantees a priori bounds on first difference quotients of solutions to (3) which are independent of the step size h. This step size independence is an important property regarding the construction of convergence theorems for solutions to the discrete problem. Our “discrete Hartman condition” appears to be less restrictive than the one given in [4]. Compatible boundary conditions, originally due to Thompson for the continuous case [7] (see also [4]), are discussed in Section 4. An existence result is presented in Section 5. If f satisfies some simple differential inequalities and a growth condition, and furthermore, if the boundary conditions are strongly compatible, then for sufficiently small step size problem (3),(4) h as a solution. Our existence results improve those in [4] and we provide an example showing this. In Section 6, we apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.