Several Proofs

Shao-Yuan Huang and Shin-Hwa Wang

In this website, we apply fundamental mathematical techniques to prove (3.56) and (3.57) in proof of Lemma 3.4(i), and to explain how to compute Lemma 3.4(ii)-(iv), appeared in our joint paper:

A variational property on the evolutionary bifurcation curves for the one-dimensional perturbed Gelfand problem from combustion theory

We remark that all computations of these proofs are supported by symbolic manipulator Maple 16.

(1). We prove (3.56):

for if .

See the PROOF.

(2). We prove (3.57):

for if .

See the PROOF.

Next, we explain how to compute Lemma 3.4(ii)-(iv).

(3). Show how to compute the integral

.

See the Link.

(4). Show how to compute the integral

.

See the Link.

(5). Show how to compute the integral

.

See the Link.