Derive from the 4.93 equation to the 4.94 equation A simple implicit scheme for the heat equation was proposed by Laasonen (1949). Thealgorithm for this scheme isu+¹-uAtUt αuxx =u+1-2u+¹+₁+1= α-If we make use of the central-difference operator-we can rewrite Equation 4.92 in the simpler form:u₁¹ujAt8u =u+1-2u+u₁_₁(Ax)²a(Ax)²12= α8u+1(4x)²This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionallystable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linearalgebraic equations must be solved at each time level n +1.The modified equation for this scheme isUxxxx+ [ { α² (At)² + 1 2 α²³At(^x)² -+12___a (4x) *] ¹.*360(4.92)+..(4.93)(4.94)

It is asked to prove the derivation

A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At α u-2u+¹+u (Ax)² If we make use of the central-difference operator 8uu1-2u+u₁-1 we can rewrite Equation 4.92 in the simpler form: u-u At - ) = [/2a²4² + ut – Qlixx ² At+ α(4x)² 12 8u+1 (Ax)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is + [10²³(At)² + 1 20²A² (Ax)² + 360° 300 (4x)* Ju .... (4.92) *** (4.93) (4.94)

A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At α u-2u+¹+u (Ax)² If we make use of the central-difference operator 8uu1-2u+u₁-1 we can rewrite Equation 4.92 in the simpler form: u-u At - ) = [/2a²4² + ut – Qlixx ² At+ α(4x)² 12 8u+1 (Ax)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is + [10²³(At)² + 1 20²A² (Ax)² + 360° 300 (4x)* Ju .... (4.92) *** (4.93) (4.94)

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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