Thanks for the formatting tip, Joe. The mathematical problem I am trying to solve is like the conformal map: w=sin(z) which gives a pair of conjugate functions either of which can be considered to be a potential function. This function is bounded by an ellipse but is otherwise the the solution I am looking for. I am setting up the problem with the varied conditions on one side in a way that reflects this mapping - the straight edge is really two distinct boundaries that have becom remapped. There may be an analytical solution to this problem but I think I really need know how to do it numerically as I also want to look at 'untidy' variations on the boundary conditions. The problem is quite easy to solve numerically - although I have used a rectangular grid, and not in Maple - but it is not then easy to get to a usable approximation of the potential surface. I used bicubic spline surface fit which gave awkward edges with my cartesian coords. I then got gradients from the fit and integrated to get the contour lines of the potential function which is what I am really after. It is very messy and I was hoping that Maple may have built-in procedures to ease the process. I am licking my wounds at the moment and looking around for another line of attack. Thanks again. JB

Thanks for the formatting tip, Joe. The mathematical problem I am trying to solve is like the conformal map: w=sin(z) which gives a pair of conjugate functions either of which can be considered to be a potential function. This function is bounded by an ellipse but is otherwise the the solution I am looking for. I am setting up the problem with the varied conditions on one side in a way that reflects this mapping - the straight edge is really two distinct boundaries that have becom remapped. There may be an analytical solution to this problem but I think I really need know how to do it numerically as I also want to look at 'untidy' variations on the boundary conditions. The problem is quite easy to solve numerically - although I have used a rectangular grid, and not in Maple - but it is not then easy to get to a usable approximation of the potential surface. I used bicubic spline surface fit which gave awkward edges with my cartesian coords. I then got gradients from the fit and integrated to get the contour lines of the potential function which is what I am really after. It is very messy and I was hoping that Maple may have built-in procedures to ease the process. I am licking my wounds at the moment and looking around for another line of attack. Thanks again. JB

I'm afraid that solution doesn't satisfy the 4th constraint. (My last post was truncated - so it wasn't given there.) It is du/d_theta=0 at theta=0 and r<0.5. The solution will be a little like the complex map w=sin(z) but will sit in a circle rather than an ellipse. However I am actually trying to find the numerical solution as the next step is to find solutions to other problms - similar but not so neat - that are unlikely to have analytical solutions. My ultimate goal is the curves fitted to the contours of this potential function, and to its conjugate harmonic function, but the plan is beginning to seem increasingly unrealistic. I solved this in Mathcad using a rectangular mesh, fitting a bicubic spline function to the solution and integrating along contours. Presumably I could do the same in Maple but I was hoping that there would be superior power in Maple. Thanks for your help here. John Brew

I'm afraid that solution doesn't satisfy the 4th constraint. (My last post was truncated - so it wasn't given there.) It is du/d_theta=0 at theta=0 and r<0.5. The solution will be a little like the complex map w=sin(z) but will sit in a circle rather than an ellipse. However I am actually trying to find the numerical solution as the next step is to find solutions to other problms - similar but not so neat - that are unlikely to have analytical solutions. My ultimate goal is the curves fitted to the contours of this potential function, and to its conjugate harmonic function, but the plan is beginning to seem increasingly unrealistic. I solved this in Mathcad using a rectangular mesh, fitting a bicubic spline function to the solution and integrating along contours. Presumably I could do the same in Maple but I was hoping that there would be superior power in Maple. Thanks for your help here. John Brew

I have mixed BCs. I am trying to find a potential function in a quadrant. (Whole circle really but I am using symmetry.) The BCs are, considering the 1st quadrant: 1) The potential is 0 at the left-hand radius (all r, theta=pi/2). 2) The potential is 1 at the right-hand end of the bottom radius (r>1/2 and theta=0). 3) The BC is of Neumann type around the curved edge of the quadrant (r=1, all theta), and 4) at also the left-hand end of the bottom radius (r<0.5 and theta=0). I expect the contours of the solution to be the upper halves of hyperbolae (or something like). The solution will be very like the conformal map using a w=sin(z) mapping except that this sits inside an ellipse rather than a circle. I am feeling increasingly uneasy about my choice of Maple's pdsolve to get a numeric solution to this since it seems to be explicitly geared to time-dependent problems. My actual interest is a bit further than this solution. Later I want to look at other shapes (rather than circles) that may be not so neat, and also other definitions of the potential positions. Consequently I am not looking for an analytical solution for this case as they are unlikely to exist for the more complex cases I will be looking for later. I was hoping to get a numerical solution and then find something in Maple that would allow me generate contour lines. (An easier way than by integrating a pde.) In fact I am also looking for the conjugate harmonic function to the solution which I can find by switching the 'neumann' and the 'potential' boundary conditions. I am beginning to feel I should have tried something easier as an intro to Maple. Thanks again, John

I have mixed BCs. I am trying to find a potential function in a quadrant. (Whole circle really but I am using symmetry.) The BCs are, considering the 1st quadrant: 1) The potential is 0 at the left-hand radius (all r, theta=pi/2). 2) The potential is 1 at the right-hand end of the bottom radius (r>1/2 and theta=0). 3) The BC is of Neumann type around the curved edge of the quadrant (r=1, all theta), and 4) at also the left-hand end of the bottom radius (r<0.5 and theta=0). I expect the contours of the solution to be the upper halves of hyperbolae (or something like). The solution will be very like the conformal map using a w=sin(z) mapping except that this sits inside an ellipse rather than a circle. I am feeling increasingly uneasy about my choice of Maple's pdsolve to get a numeric solution to this since it seems to be explicitly geared to time-dependent problems. My actual interest is a bit further than this solution. Later I want to look at other shapes (rather than circles) that may be not so neat, and also other definitions of the potential positions. Consequently I am not looking for an analytical solution for this case as they are unlikely to exist for the more complex cases I will be looking for later. I was hoping to get a numerical solution and then find something in Maple that would allow me generate contour lines. (An easier way than by integrating a pde.) In fact I am also looking for the conjugate harmonic function to the solution which I can find by switching the 'neumann' and the 'potential' boundary conditions. I am beginning to feel I should have tried something easier as an intro to Maple. Thanks again, John

You may well be right. I thought I had effectively 2 ranges for r. I can add an extra theta point (zero at r=0) from symmetry conditions. I could also halve the problem (semi-circle by symmetry) and get rid of the s third r condition - which is a duplicate because I have a wrap around at theta = 0 and 2*pi. It sounds as if the boundary conditions are not really designed to cope with polar coordinates - or perhaps it is my (total) unfamiliarity with Maple that is the problem. I will look out the book. It sounds as if it would be a big help.

You may well be right. I thought I had effectively 2 ranges for r. I can add an extra theta point (zero at r=0) from symmetry conditions. I could also halve the problem (semi-circle by symmetry) and get rid of the s third r condition - which is a duplicate because I have a wrap around at theta = 0 and 2*pi. It sounds as if the boundary conditions are not really designed to cope with polar coordinates - or perhaps it is my (total) unfamiliarity with Maple that is the problem. I will look out the book. It sounds as if it would be a big help.

Thanks for the tip, but this seems to apply to variables of a single function only. However I have now got Maple to accept conditions defined on a range using a range variable: > bc := {u(.5 .. 1, Pi) = -1, (D[1](u))(1, theta) = 0, u(.5 .. 1, 0) = 1, u(.5 .. 1, 2*Pi) = 1}; which it accepts (although I am not sure whether it has interpreted it as I hope) but I now get an error message: Error, (in pdsolve/numeric/par_hyp) Incorrect number of boundary conditions, expected 0, got 2 I am now confused about the whole pde solution system which seems to require one of the variables to be time. I just have two space variables. I shall return to the Help pages. Thanks again. John Brew

Thanks for the tip, but this seems to apply to variables of a single function only. However I have now got Maple to accept conditions defined on a range using a range variable: > bc := {u(.5 .. 1, Pi) = -1, (D[1](u))(1, theta) = 0, u(.5 .. 1, 0) = 1, u(.5 .. 1, 2*Pi) = 1}; which it accepts (although I am not sure whether it has interpreted it as I hope) but I now get an error message: Error, (in pdsolve/numeric/par_hyp) Incorrect number of boundary conditions, expected 0, got 2 I am now confused about the whole pde solution system which seems to require one of the variables to be time. I just have two space variables. I shall return to the Help pages. Thanks again. John Brew