We suggest a novel iterative scheme for solutions of singular boundary value problems (SBVPs) that is obtained by embedding Green's function into the Picard-Mann Hybrid (PMH) iterative scheme. This new scheme we call PMH-Green's iterative scheme and prove its convergence towards a sought solution of certain SBVPs. We impose possible mild conditions on the operator or on the parameters involved in our scheme to obtain our main outcome. After this, we prove that this new iterative scheme is weak $ w^{2} $-stable. Eventually, using two different numerical examples of SBVPs, we show that our new approach suggests highly accurate numerical solutions as compared the corresponding Picard-Green's and Mann-Green's iterative schemes.
Citation: Junaid Ahmad, Muhammad Arshad, Reny George. A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs[J]. AIMS Mathematics, 2023, 8(12): 29517-29534. doi: 10.3934/math.20231511
We suggest a novel iterative scheme for solutions of singular boundary value problems (SBVPs) that is obtained by embedding Green's function into the Picard-Mann Hybrid (PMH) iterative scheme. This new scheme we call PMH-Green's iterative scheme and prove its convergence towards a sought solution of certain SBVPs. We impose possible mild conditions on the operator or on the parameters involved in our scheme to obtain our main outcome. After this, we prove that this new iterative scheme is weak $ w^{2} $-stable. Eventually, using two different numerical examples of SBVPs, we show that our new approach suggests highly accurate numerical solutions as compared the corresponding Picard-Green's and Mann-Green's iterative schemes.
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Junaid Ahmad, Muhammad Arshad, Reny George. A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs[J]. AIMS Mathematics, 2023, 8(12): 29517-29534. doi: 10.3934/math.20231511
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