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solving fourth order ordinary K242/44

differential equation using adomian

decomposition method and variational

iteration method

NAME : SITI NOR BAHIYAH BINTI MOHD SALLEH

SUPERVISOR : W.KHAIRIYAH HULAINI BINTI WAN RAMLI

ABSTRACT PROBLEM STATEMENT

This study addresses the challenge of solving fourth-order Solving higher-order ODEs analytically is difficult, especially

Ordinary Differential Equations (ODEs), particularly nonlinear when they involve nonlinear terms, making exact solutions

types, by using two semi-analytical methods: Adomian rare or complex.

Decomposition Method (ADM) and Variational Iteration Method Adomian Decomposition Method (ADM) simplifies nonlinear

(VIM). Using MAPLE software, both methods showed close problems using Adomian polynomials but may require

approximations to the exact solution with minimal errors. The heavy computation and multiple iterations.

research highlights the effectiveness and convergence of these Variational Iteration Method (VIM) offers a simpler

methods for solving higher-order ODEs alternative by using correction functionals, but its

effectiveness depends on correctly selecting the Lagrange

OBJECTIVES multiplier.

To solve a fourth-order ordinary differential equation using the Adomian This study aims to compare ADM and VIM in solving fourth-

Decomposition Method.

order ODEs, focusing on their accuracy, convergence, and

To solve a fourth-order ordinary differential equation using the Variational practical efficiency.

Iteration Method.

To compare the error between exact solution and approximate solution.
RECOMMENDATION

Future research should explore hybrid

METHOD & IMPLEMENTATION methods that combine ADM and VIM for

improved accuracy and efficiency.

ADOMIAN DECOMPOSITION METHOD VARIATIONAL ITERATION METHOD
Refining the selection of Lagrange

Rewrite the ODE in operator form Rewrite the ODE in operator form multipliers in VIM and testing on more

complex or real-world equations is

encouraged. Additionally, applying these

Where L is the highest order derivative methods to a wider range of examples and

Construct correction functional improving automation in computations

Apply inverse operator to isolate u(x)

would enhance their practical use.

CONCLUSION

Determine the using initial condition and

inhomogenous term Find the Lagrange multiplier λ(s) This study successfully applied the

Adomian Decomposition Method (ADM)

and Variational Iteration Method (VIM) to

solve fourth-order ordinary differential

Decompose nonlinear terms (if any) into Choose initial approximation using initial equations. Both methods produced highly

Adomian polynomials. conditions. accurate results with minimal error. ADM

Generate next terms using recursion Compute successive approximations showed slightly better precision, while

VIM was simpler and faster due to

Final solution is taken from the last approximation avoiding Adomian polynomial

which is iteration n=10 calculations. Overall, both are reliable and

Sum the series up to desired iteration (e.g. effective, and the choice depends on the

10th) for the approximate solution. problem type and computational

preference.

RESULT AND DISCUSSION

Vim Adm Adm Vim Adm Vim