==== ONE DIMENSIONAL HELMHOLTZ EQUATION DOF=2 ==== ==== DIRICHLET ------- NEUMANN PROBLEM ==== ---- GALERKINS WEIGHTING FUNCTION---- # OF GL INTEGRATION SAMPLING POINTS = 6 APPROXIMATING FUNCTION: U(X)=F0(X)+A1*F1(X)+A2*F2(X) WHERE F0(X) = U0+SL(X/L) F1(X) = (X/L)*(1-X/L) + (X/L) F2(X) = ((X/L)**K*(1-X/L))**2 LENGTH OF DOMAIN = 0.5 NUMBER OF SEGMENTS FOR INTEGRATION = 10000 DX FOR DERIVATIVE EVALUATION = 0.000005 X-COORDINATE OF LEFT END BOUNDARY = 0. X-COORDINATE OF RIGHT END BOUNDARY = 0.5 ALPHASQ = 1. U(X) AT X=0 = 1. S AT X=L = 0. A1= 0.13656633221803283 A2= 0.002926421405357634 U(X)=F0(X)+ 0.13656633221803283 *F1(X)+ 0.002926421405357634 *F2(X) X-COORDINATE U(X) DU/DX EXACT(X) |U(X)-EXACT(X)| 0. 1. 0.5462653288721313 1. 0. 0.05 1.0260532469341597 0.4956421404674958 1.0260540050078621 7.58073702478157E-7 0.1 1.0495431438126261 0.4437547380163474 1.0495434093618006 2.655491744274485E-7 0.15000000000000002 1.0704099916387302 0.39074358974459783 1.0704095017839756 4.898547545373333E-7 0.2 1.0886011148271755 0.33674916388088766 1.0886001279101831 9.869169923515386E-7 0.25 1.1040708612040382 0.28191192865033393 1.1040698206486024 0.0000010405554358161595 0.30000000000000004 1.1167806020067679 0.2263723522866852 1.1167799138238472 6.881829206495382E-7 0.35000000000000003 1.1266987318841866 0.17027090300900014 1.1266986388222686 9.30619179495551E-8 0.4 1.133800668896489 0.11374804905107865 1.1338012039969423 5.351004532805348E-7 0.45 1.1380688545152435 0.05694425864029318 1.138069856633876 0.0000010021186325204212 0.5 1.1394927536233905 0. 1.1394939273245492 0.0000011737011587076296