Concept:
Chronological arrangement requires arranging writers according to the period of their major literary contributions.
Step 1: Identify the earliest writer.
Toru Dutt belonged to the nineteenth century and is among the earliest Indian English writers.
\[
E \rightarrow \text{First}
\]
Step 2: Place Rabindranath Tagore.
Rabindranath Tagore's major literary works appeared after Toru Dutt.
\[
C \rightarrow \text{Second}
\]
Step 3: Place Sri Aurobindo.
Sri Aurobindo contributed significantly during the late nineteenth and early twentieth centuries.
\[
A \rightarrow \text{Third}
\]
Step 4: Place R.K. Narayan.
R.K. Narayan belongs mainly to twentieth-century Indian English fiction.
\[
B \rightarrow \text{Fourth}
\]
Step 5: Place Amitav Ghosh.
Amitav Ghosh is a contemporary Indian English writer.
\[
D \rightarrow \text{Fifth}
\]
Step 6: Write the final order.
\[
E,\; C,\; A,\; B,\; D
\]
Step 7: Conclusion.
Thus, the correct answer is:
\[
\boxed{E,\; C,\; A,\; B,\; D}
\]