I am looking for a new iterative method or analytic-approximate method to solve ODE\PDE not the already known methods such as:ADM, VIM, HAM, HPM, DJM, DTM,,,,,but new method.
Best regards,
You can create variants of the known methods and attempt using your newly created variant to solve the problem. Maybe, you would achieve a breakthrough.
You can try these two methods:
1. Rapidly convergent approximation analytic method (RCAM) is an iterative analytic method for obtaining exact/ approximate solutions of nonlinear differential equations. Few references are as follows:
i. A Rapidly Convergent Approximation Method for Nonlinear Ordinary Differential Equations, Prakash, Kumar Das, M.M Panja , IJSEAS 2 (8), 334-348
ii. Solutions and conserved quantities of Biswas–Milovic equation by using the rapidly convergent approximation method, PK Das, D Singh, MM Panja, Optik 174, 433-446
iii. Rapidly Convergent Approximation Method to Chiral Nonlinear Schrodinger’s Equation in (1+2)-dimensions, PK Das, Sohag Journal of Mathematics 5 (1), 29-33.
iv. Piecewise smooth localized solutions of Liénard‐type equations with application to NLSE, PK Das, S Mandal, MM Panja, Mathematical Methods in the Applied Sciences 41 (17), 7869-7887
or you can try,
2. Piecewise Analytic Method (PAM) is based on dividing the solution interval into subintervals and obtaining an approximate analytic solution which is very accurate and can be applied to each subinterval successively. Few references are as follows:
i. T. A. Abassy, Piecewise Analytic Method. DOI:10.14419/ijamr.v1i1.22
https://www.researchgate.net/publication/265162801_Piecewise_Analytic_Method
ii. T. A. Abassy, Introdction to Piecewise Analytic Method. Vol. 3(S). July, 11, 2012, No. 10, pp. 1‐19. https://www.researchgate.net/publication/265162885_INTRODUCTION_TO_PIECEWISE_ANALYTIC_METHOD
iii. T. A. Abassy, Solving nonlinear 2nd order differential equations using piecewise analytic method(Pendulum Equations), DOI: 10.13140/RG.2.2.19063.88489
https://www.researchgate.net/publication/331023867_Solving_Nonlinear_2_nd_Order_Differential_Equations_Using_Piecewise_Analytic_Method_Pendulum_Equations
Collocation methods can be useful for solving partial differential equations.
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