−u′′(x)+u(x)=k.x, x∈(0,1), k∈ u(0) = 0, u(1)= 0
(The numerical value of k is the last three digits of your school number) k=910
Solve the boundary value problem given above by using equal lenght finite elements with 3 finite elements and draw the exact solution and the finite element solution of the problem in the same graph.
in Other Math Topics by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Rewrite the DE:

u''(x)-u(x)=-910x.

The characteristic equation is found by solving r²-1=0. (r represents u'').

So r=±1, giving us u₁=Ae⁻ˣ+Beˣ, where A and B will be found later from the initial conditions.

The particular equation we can guess has the form u₂=ax+b, where we can work out  constants a and b:

u₂'=a and u₂''=0, so the DE becomes:

0-ax-b=-910x, from which b=0 and a=910, making u₂=910x.

The complete solution is u₁+u₂=u=Ae⁻ˣ+Beˣ+910x.

But u(0)=0=A+B, so B=-A.

u(1)=A/e-Ae+910=0, A-Ae²+910e=0.

From this A=910e/(e²-1)=387.1677 approx.

Therefore u=387.1677e⁻ˣ-387.1677eˣ+910x. In the graph below you only need the “hump” between x=0 and x=1. This will be the region divided into segments of equal length.

 

Another way if writing this is:

u(x)=910x-(910e/(e²-1))sinh(x).

[sinh(x)=½(eˣ-e⁻ˣ)]

Now, using the finite element method, we get:

d²u/dx²-u=-910x.

Divide (0,1) into n equal segments, so each segment has length=1/n. Call this length ∆x. The division gives rise to nodes numbered 1 to n+1.

If n=3, so ∆x=⅓ and 1≤i≤4, where i is the node number. The value of x at each node is xᵢ₋₁ and the corresponding u value is uᵢ₋₁. This means that the first node is at x=0 and the sixth node is at x=1. 

x₀=0, x₁=⅓, x₂=⅔, x₃=1.

Now consider the series:

(x₀,u₀), (x₁,u₁), (x₂,u₂), (x₃,u₃).

Since xᵢ₊₁-xᵢ=∆x, the first differences (differential or derivative) are:

(u₁-u₀)/∆x, (u₂-u₁)/∆x, (u₃-u₂)/∆x.

The second differences are:

(u₂-u₁-u₁+u₀)/(∆x)²=(u₂-2u₁+u₀)/(∆x)²,

(u₃-2u₂+u₁)/(∆x)².

From which we can make the approximation:

d²u/dx²≈(uᵢ₊₁-2uᵢ+uᵢ₋₁)/(∆x)².

The boundary conditions are u₀=0 and u₅=0 (because u(0)=u(1)=0).

The DE can now be written:

(uᵢ₊₁-2uᵢ+uᵢ₋₁)/(∆x)²-uᵢ=-910xᵢ, or 9uᵢ₊₁-19uᵢ+9uᵢ₋₁=-910xᵢ, since (∆x)²=1/9.

Therefore, we arrive at a system of 2 equations:

9u₂-19u₁=-910/3, so u₂=(19u₁-910/3)/9;

-19u₂+9u₁=-1820/3, so -19(19u₁-910/3)/9+9u₁=-1820/3,

-361u₁/9+17290/27+9u₁=-1820/3.

Multiplying through by 27:

-1083u₁+17290+243u₁=-16380,

-840u₁+33670=0, u₁=33670/840=481/12=40.083.

u₂=611/12=50.917.

So the points to plot are A(⅓,481/12) and B(⅔,611/12).

These compare with u(⅓)=40.415 and u(⅔)=51.345.

 

 

by Top Rated User (1.1m points)

Related questions

1 answer
asked Feb 18, 2020 in Other Math Topics by anonymous | 190 views
1 answer
1 answer
1 answer
asked Sep 20, 2021 in Other Math Topics by anonymous | 853 views
1 answer
asked Apr 5, 2021 in Other Math Topics by Thatsd12 | 352 views
2 answers
asked Nov 29, 2020 in Other Math Topics by Rapht | 5.3k views
1 answer
asked Dec 5, 2014 in Other Math Topics by ruth | 9.0k views
1 answer
1 answer
1 answer
asked Sep 15, 2020 in Algebra 1 Answers by anonymous | 548 views
1 answer
asked May 7, 2019 by anonymous | 744 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users