In the Previous Three Problems, the Solution of the Original $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) t }$

question 13

Multiple Choice

In the previous three problems, the solution of the original problem is

• A) \(u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) t }\) , where \(c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x\)
• B) \(u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \cos ( n \pi t /L )\) , where \(c _ { 0 } = \int _ { 0 } ^ { L } f ( x ) d x / L , c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L\)
• C) \(u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sin ( n \pi t / L )\) , where \(c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L\)
• D) \(u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - nxt / L }\) , where \(c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L\)
• E) \(u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - n x t / L }\) , where \(c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x\)

---

Definitions:

Memory Construction

The process by which memories are formed, consolidated, and potentially distorted over time, involving both encoding and retrieval stages.

Proactive Interference

A phenomenon where older memories hinder the recall of newly learned information.

Memory Construction

The process of creating, consolidating, and encoding memories based on experiences, influenced by various factors including perception and social inputs.

Proactive Interference

The phenomenon where old memories hinder the recall of newer memories.