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I have a partial differential equation to solve, and all I would like to find out is the step by step process of going about it, so I can work it out myself. We have u*u_x+u_t = 2, where u_x is the partial derivative of u with respect to x, and such. The initial conditions are given as u(x, 0 ) = -x. We are first asked to find the Characteristic curves. What I did was use the chain rule, and went about as follows : To begin with, let us look at the Chain Rule to see if we can make the equation above fit the pattern it gives. For this we have : du/dt=δu/δx*dx/dt + δu/δt*dt/dt, and so, to give it a solution more conducive to this, we make sure that the u in the problem is analogous to the dx/dt in this variation of the Transport Convection Equation, both being the coefficients of δu/δx , and the 2 from the Right Hand Side of the problem is like the du/dt in the Chain Rule expression, and it is obvious that dt/dt=1.
So dx/dt*δu/δx + δu/δt = du/dt compared to u*δu/δx + δu/δt=2 makes u=dx/dt, and du/dt=2. For a start, this means that along the characteristic lines, u changes for each second with a slope of two, where if we integrate du/dt=2, and get ∫(du/dt) dt=∫2dt, we end up with u=2t+A , with A some constant, or function not dependent upon t, and at t=0, because the initial condition states that u(x,0)= -x, then A= -x, while for u=dx/dt, this implies that x=ut+B, with B a constant of integration or function not depending on t, since dx/dt= d/dt(ut)=u. I also went on to add : From the information above, since u=2t+A,∀t∈R∶t ≥0, and it was found that A= -x, then u=2t-x. But since it is also known that x=ut+B, and if we rearrange u=2t-x, we can also shew that x=2t-u, then ut+B= 2t-u.
Now out of this, which are the characteristic curves ? Are they the lines x=ut+B, and u=2t-x, or have I totally mucked this up ? Should I have instead used the Lagrange Charpit equations, or a change of variables with t=tau and x=xi ? We are required next to find the solution u in terms of the original variables x and t, so I guess I should change variables, but then I am fuzzy on how to relate this to the chain rule equation. We are also to go on to sketch this, and I assume these are lines in the plane. We are also asked to Consider this same initial value problem but with a new initial condition that u(x, 0) = f(x), where f_0(x) is bounded, then to find those characteristic curves and from the Jacobian, determine the earliest possible time at which the solution will break down. We are also informed that it is not possible to find an explicit form of u as a function of x and t but it is possible to find an implicit solution F(x, t, u) = 0, and we are asked to find it, and I have been looking at YouTube videos all weekend to work out the procedures for solving PDE'S like these, but I am still not sure how to proceed. Finally, we are asked to consider the initial value problem u_t - u^2*u_x = 0, with initial conditions u(x, 0) = g(x) = { negative half, if x less than or equal to zero ; 1 if x between zero and 1 exclusive ; and one half when x is greater than or equal to one } From this we are asked to sketch or plot those base characteristics, then find the solution, as well as determine in which region it is single-valued. Next we sketch or plot the solution for t = 0, 2, 4, and finally find the shock solution for this system.
I have some ideas, but I just need some hints to put it all together. Thank You Chris the Russian Christopher Lilly 08:01, 3 April 2017 (UTC).
Yes, it is non linear because of the u term as a coefficient of the partial derivative of u with respect to x, and Thank You. What I did next was to make the Chain Rule, where du/dtau=δu/δx*dx/dtau + δu/δt*dt/dtau, then I got u=dx/dtau, 2=du/dtau, and dt/dtau=1, since these matched up with their analogous coefficients in the original equation, then I said because u=dx/dτ, ∫udτ= ∫dx/dτ dτ → x= uτ+ ξ → ξ = x-uτ, and we also let t=τ . In addition, since du/dτ=2, then ∫du/dτ dτ =∫2dτ → u=2τ+ ξ → ξ = u-2τ, but then I have two expressions for xi ( ξ ), where one equals x-uτ, and the other is u-2τ, yet I am not sure if that is right, or what to do next. I need to find the characteristic curves, then solve to find u(x, t ).. Chris the Russian Christopher Lilly 01:48, 5 April 2017 (UTC)
Okay, I shall look at that and try to put the question again, using that to make the derivatives clearer, thank You. Chris the Russian Christopher Lilly 03:01, 5 April 2017 (UTC)
How much is ? (t, t0 both nonnegative reals)
On the one hand, WolframAlpha seems to think that .
On the other hand, when I do the computations by hand (develop the product, change variables), I end up finding (EDIT 07:22, 4 April 2017 (UTC): arcsin(1) is pi/2, not pi/4...)
Both expressions look compatible on limit cases, but WA's numerical evaluation tells me they are not equal ( example with t0=1). Tigraan Click here to contact me 16:33, 3 April 2017 (UTC)
In the special case where , this is equivalent to which has value , which shows that @ Tigraan:'s value is incorrect.-- Jasper Deng (talk) 20:43, 3 April 2017 (UTC)
If I have a set of correlated numeric attributes for objects, GCI claims that the probability of an object having all attributes in the 95% range per attribute is equal or greater than the product of each individual probability being in the 95% range for each individual attribute. Is it valid to reverse this? May I make a valid claim that the probability of being outside the 95% for a specific attribute is lowered if the object is within the 95% range for another correlated attribute? Specific application: If a patient is deemed healthy based on X measurable attributes, the probability that the patient will be unhealthy for another attribute is less than the probability of being unhealthy based solely on that attribute. 209.149.113.5 ( talk) 17:33, 3 April 2017 (UTC)
By brute force if that's the best they can do. How powerful a general purpose computer is needed to solve a random quintic equation in under 1 hour (worst case) if the equation is picked from the ones with only integers below z-digits? Would increasing z by 1 take 10 hours or 105 or what? Is this a very parallelizable problem? If under z-digits takes 1 hour how long would sexic equations under z-digits take? How many digits for a quartic, cubic, quadratic or linear equation before a modern PC starts slowing down? (must be a lot?) Sagittarian Milky Way ( talk) 18:23, 3 April 2017 (UTC)
An ARIMA(1,0,0) model can be expressed in the matrix as follows:
_ _ _ _ | y_{1} | | 1 0| | y_{2} | |1-\phi=0.68 0| | ... | |... 0| _ _ | y_{n_{1}} | |.68 0| |L | | --- | = |--- ---| * | | | y_{n_1}+1} | |.68 1| |\delta| | y_{n_{1}+2}| |.68 1| _ _ | ... | |... ...| | y_{N} | |.68 1| _ _ _ _
Therefore,
_ _ |L | | | = (((X^T)*X)^-1)*((X^T)Y) |\delta| _ _|
Where X is the Nx2 matrix in the first matrix form above (I goofed around with Latex for too long, pardon my ascii math).
So, doing the math (with software),
(((X^T)*X)^-1)*((X^T)Y)=0.56.
Question: how do I differentiate L and \delta? Schyler ( exquirere bonum ipsum) 21:42, 3 April 2017 (UTC)
Mathematics desk | ||
---|---|---|
< April 2 | << Mar | April | May >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I have a partial differential equation to solve, and all I would like to find out is the step by step process of going about it, so I can work it out myself. We have u*u_x+u_t = 2, where u_x is the partial derivative of u with respect to x, and such. The initial conditions are given as u(x, 0 ) = -x. We are first asked to find the Characteristic curves. What I did was use the chain rule, and went about as follows : To begin with, let us look at the Chain Rule to see if we can make the equation above fit the pattern it gives. For this we have : du/dt=δu/δx*dx/dt + δu/δt*dt/dt, and so, to give it a solution more conducive to this, we make sure that the u in the problem is analogous to the dx/dt in this variation of the Transport Convection Equation, both being the coefficients of δu/δx , and the 2 from the Right Hand Side of the problem is like the du/dt in the Chain Rule expression, and it is obvious that dt/dt=1.
So dx/dt*δu/δx + δu/δt = du/dt compared to u*δu/δx + δu/δt=2 makes u=dx/dt, and du/dt=2. For a start, this means that along the characteristic lines, u changes for each second with a slope of two, where if we integrate du/dt=2, and get ∫(du/dt) dt=∫2dt, we end up with u=2t+A , with A some constant, or function not dependent upon t, and at t=0, because the initial condition states that u(x,0)= -x, then A= -x, while for u=dx/dt, this implies that x=ut+B, with B a constant of integration or function not depending on t, since dx/dt= d/dt(ut)=u. I also went on to add : From the information above, since u=2t+A,∀t∈R∶t ≥0, and it was found that A= -x, then u=2t-x. But since it is also known that x=ut+B, and if we rearrange u=2t-x, we can also shew that x=2t-u, then ut+B= 2t-u.
Now out of this, which are the characteristic curves ? Are they the lines x=ut+B, and u=2t-x, or have I totally mucked this up ? Should I have instead used the Lagrange Charpit equations, or a change of variables with t=tau and x=xi ? We are required next to find the solution u in terms of the original variables x and t, so I guess I should change variables, but then I am fuzzy on how to relate this to the chain rule equation. We are also to go on to sketch this, and I assume these are lines in the plane. We are also asked to Consider this same initial value problem but with a new initial condition that u(x, 0) = f(x), where f_0(x) is bounded, then to find those characteristic curves and from the Jacobian, determine the earliest possible time at which the solution will break down. We are also informed that it is not possible to find an explicit form of u as a function of x and t but it is possible to find an implicit solution F(x, t, u) = 0, and we are asked to find it, and I have been looking at YouTube videos all weekend to work out the procedures for solving PDE'S like these, but I am still not sure how to proceed. Finally, we are asked to consider the initial value problem u_t - u^2*u_x = 0, with initial conditions u(x, 0) = g(x) = { negative half, if x less than or equal to zero ; 1 if x between zero and 1 exclusive ; and one half when x is greater than or equal to one } From this we are asked to sketch or plot those base characteristics, then find the solution, as well as determine in which region it is single-valued. Next we sketch or plot the solution for t = 0, 2, 4, and finally find the shock solution for this system.
I have some ideas, but I just need some hints to put it all together. Thank You Chris the Russian Christopher Lilly 08:01, 3 April 2017 (UTC).
Yes, it is non linear because of the u term as a coefficient of the partial derivative of u with respect to x, and Thank You. What I did next was to make the Chain Rule, where du/dtau=δu/δx*dx/dtau + δu/δt*dt/dtau, then I got u=dx/dtau, 2=du/dtau, and dt/dtau=1, since these matched up with their analogous coefficients in the original equation, then I said because u=dx/dτ, ∫udτ= ∫dx/dτ dτ → x= uτ+ ξ → ξ = x-uτ, and we also let t=τ . In addition, since du/dτ=2, then ∫du/dτ dτ =∫2dτ → u=2τ+ ξ → ξ = u-2τ, but then I have two expressions for xi ( ξ ), where one equals x-uτ, and the other is u-2τ, yet I am not sure if that is right, or what to do next. I need to find the characteristic curves, then solve to find u(x, t ).. Chris the Russian Christopher Lilly 01:48, 5 April 2017 (UTC)
Okay, I shall look at that and try to put the question again, using that to make the derivatives clearer, thank You. Chris the Russian Christopher Lilly 03:01, 5 April 2017 (UTC)
How much is ? (t, t0 both nonnegative reals)
On the one hand, WolframAlpha seems to think that .
On the other hand, when I do the computations by hand (develop the product, change variables), I end up finding (EDIT 07:22, 4 April 2017 (UTC): arcsin(1) is pi/2, not pi/4...)
Both expressions look compatible on limit cases, but WA's numerical evaluation tells me they are not equal ( example with t0=1). Tigraan Click here to contact me 16:33, 3 April 2017 (UTC)
In the special case where , this is equivalent to which has value , which shows that @ Tigraan:'s value is incorrect.-- Jasper Deng (talk) 20:43, 3 April 2017 (UTC)
If I have a set of correlated numeric attributes for objects, GCI claims that the probability of an object having all attributes in the 95% range per attribute is equal or greater than the product of each individual probability being in the 95% range for each individual attribute. Is it valid to reverse this? May I make a valid claim that the probability of being outside the 95% for a specific attribute is lowered if the object is within the 95% range for another correlated attribute? Specific application: If a patient is deemed healthy based on X measurable attributes, the probability that the patient will be unhealthy for another attribute is less than the probability of being unhealthy based solely on that attribute. 209.149.113.5 ( talk) 17:33, 3 April 2017 (UTC)
By brute force if that's the best they can do. How powerful a general purpose computer is needed to solve a random quintic equation in under 1 hour (worst case) if the equation is picked from the ones with only integers below z-digits? Would increasing z by 1 take 10 hours or 105 or what? Is this a very parallelizable problem? If under z-digits takes 1 hour how long would sexic equations under z-digits take? How many digits for a quartic, cubic, quadratic or linear equation before a modern PC starts slowing down? (must be a lot?) Sagittarian Milky Way ( talk) 18:23, 3 April 2017 (UTC)
An ARIMA(1,0,0) model can be expressed in the matrix as follows:
_ _ _ _ | y_{1} | | 1 0| | y_{2} | |1-\phi=0.68 0| | ... | |... 0| _ _ | y_{n_{1}} | |.68 0| |L | | --- | = |--- ---| * | | | y_{n_1}+1} | |.68 1| |\delta| | y_{n_{1}+2}| |.68 1| _ _ | ... | |... ...| | y_{N} | |.68 1| _ _ _ _
Therefore,
_ _ |L | | | = (((X^T)*X)^-1)*((X^T)Y) |\delta| _ _|
Where X is the Nx2 matrix in the first matrix form above (I goofed around with Latex for too long, pardon my ascii math).
So, doing the math (with software),
(((X^T)*X)^-1)*((X^T)Y)=0.56.
Question: how do I differentiate L and \delta? Schyler ( exquirere bonum ipsum) 21:42, 3 April 2017 (UTC)