How Do You Determine if an Ordered Pair is a Solution to an Equation?
Apr 29, · Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Answer: If a solution results in zero when subsitituted into the denominator of the equation, the solution is extraneous. This is because you obtained the solution from a simplified version of the original equation, but you have to check if the solution obtained is a real solution of the original equation.
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Then check your solution. Solving Equations of the Form ax = b. We can see that using the multiplication principle to multiply each side of an equation by 1/2 is the same as dividing each side of the equation by 2. Thus, it would seem that the multiplication principle would allow us to divide each side of the equation by any nonzero real number. To figure out if an ordered pair is a solution to an equation, you could perform a test. Identify the x-value in the ordered pair and plug it into the equation. When you simplify, if the y-value you get is the same as the y-value in the ordered pair, then that ordered pair is indeed a solution to the equation. Sample problem #3: Find the general solution for the differential equation ? 2 d? = sin(t + ) dt. Step 1: Integrate both sides of the equation: ? ? 2 d? = ?sin(t + ) dt > ? 3 = -cos(t + ) + C That’s how to find the general solution of differential equations! Tip: If your differential equation has a constraint, then what you need to find is a particular solution.
The heat equation is a second order partial differential equation that describes how the distribution of some quantity such as heat evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower.
It is a special case of the diffusion equation. For a function T x,y,z,t of three spatial variables x,y,z and the time variable t, the heat equation is:.
This equation describes the flow of heat in a homogeneous and isotropic medium, with T x,y,z,t being the temperature at the point x,y,z and time t. The value of alpha will affects the speed of the heat diffusion and it depends upon the material being used. The above equation is solved on a 2-dimensional plate which contains four boundaries and the value at those boundaries are.
The value of thermal diffusivity is taken as 1. The assumption made to solve above equation on a 2D plate are:. The material is homogeneous and isotropic. The alpha is not the function of temperature. The radiative and convective losses are neglected. The final solution is obtained in three way. Neglecting unsteady term in the equation.
Steady State Solution. So general 2D FDM form of the equation will be;. The above form will be modified according to the above three cases and the solution will be obtained. Case I. Solution by neglecting unsteady term. The equation will be solved with Jacobi, Gauss Siedel and Successive relaxation iterative method to check that which method is faster and how it works.
The matlab code to solve the above form is:. As we are observing the steady temperature distribution in the plate, the final solution with all the method will be visually same, but the number of iteration each method will take will be different. The above graph clearly shows that SOR method takes less number of iteration to converge than the gauss seidel and Jacobi. Consequently, the SOR takes less time followed by gauss seidel and jacobi method.
To get the solution faster, the successive relaxation factor should be optimum. In above case, the relaxation factor from has been tried and the factor 1.
There are various formulas are available for optimum relaxation factor, but in above case it has been observed that those are not giving optimum results. So, trial and error methos is best suitable here to choose the optimum relaxation factor. As per the error criteria given, the final solution will be the same, but the method to get the solution faster is also equally important.
And, SOR is the winner. Case II. Solution by considering unsteady term Explicit Method. The explicit form of the above equation is easy to form but care should be taken to get the stable solution.
The CFL condition based timstep is required to get the stable solution. As the solution for the previous time exist for time marching, this form of the solution is easy to solve. The iterative solvers are not required for this method because system of eqautions are not getting formed.
The RHS of the equation is completely known and there is only one unknown in the equation. So, the simple and short code for this method is,. The care should be taken while choosing the timestep for this method. The CFL criteria is considered while choosing timstep which states that,. The above formula will give the timstep of 0.
The decreasing timstep below this value will also give stable accurate solution but the time required to get the solution will be increased. So, we can say that CFL based timstep is optimum timstep for the explicit method.
Case III. Solution by considering unsteady term Implicit Method. The implicit method is unconditionally stable but it is difficult and time-consuming.
Application of implicit method on each node will form a system of linear equations and it needs to be solved by iterative methods. The iterative methods are discussed in the steady state solution and to maintain the consistency, same methods are used in this method. The results of three methods are also observed to be same as before, in which SOR will take least iterations followed by Gauss Siedel and Jacobi.
The CFL criteria is not required to be checked for calculating time step. But, choosing the higher timestep can lead to wrong solution. The timestep of 0.
The simulation has not been ran for fixed total time, because that will not allow us to get the solution faster. The total iterations and total number of timsteps are calculated and compared for all the three methods. The code to solve the 2D Heat equation by implicit method is;. The successive over-relaxation parameter is selected based on trial and error method, because the formulaes available in literature are not reliable. Thanks for choosing to leave a comment.
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Call Us Courses. Numerical Solution of 2D Heat equation using Matlab. Pritam updated on Jan 13, , pm IST. Comments 0. The assumption made to solve above equation on a 2D plate are: 1. Steady State Solution 2. Considering Unsteady term but solved by Explicit method. Considering Unsteady term but solved by Implicit method. Jacobi 2. Gauss Siedel 3. The temperature distribution plot with the number of iterations are; The above graph clearly shows that SOR method takes less number of iteration to converge than the gauss seidel and Jacobi.
As all cases are observed for final steady solution, the temperature plots are same. The explicit method is easy to form, but stability needs to be checked with CFL. The implicit method needs iterative solver, where explicit gives the direct solution. Implicit method is unconditionally stable but higher timstep can lead to wrong solution. Implicit methods are difficult to form but stability make it a attractive choice.
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