System of Lane-Emden equations as IVPs BVPs and Four Point BVPs Computation with Haar Wavelets
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In this work we present Haar wavelet collocation method and solve the following class of system of Lane-Emden equation defined as -(t^k_1 y'(t))'=t^-ω_1 f_1(t,y(t),z(t)), -(t^k_2 z'(t))'=t^-ω_2 f_2(t,y(t),z(t)), where t>0, subject to initial values, boundary values and four point boundary values: y(0)=γ_1, y'(0)=0, z(0)=γ_2, z'(0)=0, y'(0)=0, y(1)=δ_1, z'(0)=0, z(1)=δ_2, y(0)=0, y(1)=n_1z(v_1), z(0)=0, z(1)=n_2y(v_2), where n_1, n_2, v_1, v_2∈ (0,1) and k_1≥ 0, k_2≥0, ω_1<1, ω_2<1 are real constants. Results are compared with exact solutions in the case of IVP and BVP. In case of four point BVP we compare the result with other methods. Convergence of these methods is also established and found to be of second order. We observe that as resolution is increased to J=4 we get the exact values for IVPs and BVPs. For four point BVPs also at J=4, we get highly accurate solutions, e.g., the L^∞ error is of order 10^-16 or 10^-17.
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