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The following tridiagonal system must be solved as part of a larger algorithm (Crank-Nicolson) for solving partial differential equations:
Use the Thomas algorithm to obtain a solution.
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Chapter 11 Solutions
Numerical Methods For Engineers, 7 Ed
- - 7. (a) Show that x − 1 = (x − 1)(x² + x³+x²+x+1). Deduce that if @=e²i/5 then w4+w³+w²+w+1=0. (b) Let a = = 2 cos 2 and ß = 2 cos 4. Show that a = @ + 04 and ß = ²+w³. Find a quadratic equation with roots a,ß. Hence show that =(√5-1). COS 2π 5arrow_forwardConsider the system of differential equations x₁ = 10/3x₁ +4/3x2 x2 = 8/3x1 +14/3x2 9 where 1 and 2 are functions of t. Our goal is first to find the general solution of this system and then a particular solution.arrow_forwardWrite the system of differential equations given in the 1st photo in the normal form defined in the 2nd photo and solve the resulting system by means of the eigenvalues and eigen-vectors of the square matrix A.arrow_forward
- For each of the following systems of differential equations, use the eigenvalues and eigenvectors of the coefficient matrix to find the general solution. A. B. *1-2x17x2 2x14x2 = *1 = 2x1 *2 = 5x2 - 7x3 *3 = 2x₂ - 4x3arrow_forwardConsider the non-linear system of differential equations, x₁ = x₁(x₂ − 1), x2 = − x₂(x₁ -X2(x1 + 1).arrow_forward1 Compute: Lk=0 2k Compute: - k².arrow_forward
- Express the following first-order system as a single higher-order differential equation and solve using the given initial conditions. Make sure to change the initial conditions appropriately. Evaluate the system at t = 1 and select the most approximate vector form. x₁ = -2x₁ - 2x2 x2 = −5x₁ + x2 x₁ (0) = -4, x2 (0) = 2. O x₁ (1) [x₂ (1)] 0 [2010] ] = [ O [1(1)] [x₂ (1)] (1)] = = = -18.527 75.709 -23.026 78.965 -29.545 79.812 -34.474] 86.039arrow_forwardQ.b=−1.arrow_forwardFind the general form of the solution to the following system of differential equations, x = 2x1 + x2 + x3 x = x2 – X3 x3 = x2 + X3.arrow_forward
- please solve it on paperarrow_forward9.11 Given the equations 2x, - 6x2 – x3 = -38 -3x, - x2 + 7x3 = -34 -8x, + x2 - 2x3 = -20 (a) Solve by Gauss elimination with partial pivoting. Show all steps of the computation. (b) Substitute your results into the original equations to check your answers.arrow_forwardGiven the differential equation: Choose all correct answers. A (x²-1) y" - 6x y¹ + 12y=0 D The recurrence relation is given by: (n-2)(n-3) C C n+2 (n+2) (n+3) n y=c (1+6x4+x³) +c₂(x+x² + x³) (C) y=c (1+6x²+x4) +c₁(x+x³) = The recurrence relation is given by: C n+2 = (n-3)(n-4) (n+1) (n + 2) C n n = 0,1,2,... n=0,1,2,...arrow_forward
Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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