EXISTENCE AND UNIQUENESS IN THE LINEARISED ONE AND TWO-DIMENSIONAL PROBLEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH VARIATIONAL METHOD

. The classical solution and the strong solution of a partial diﬀerential equation problem are continuously diﬀerentiable solutions. This solution has a derivative for a continuous inﬁnity level. However, not all problems of partial diﬀerential equations can be easily obtained by strong solutions. Even the existence of a solution requires in-depth investigation. The variational formulation method can qualitatively analyze a unique solution to a partial diﬀerential equation problem. This study provides an alternative method in analyzing the problem model of partial diﬀerential equations analytically. In this study, the problem of partial diﬀerential equations will be solved analytically by using the variational method and the ﬁnite element method to conﬁrm it numerically. Kata


Introduction
Partial differential equations are widely used in modelling various physical, geometric and probabilistic phenomena. The partial differential equation modelling can be in the form of linear and nonlinear equations depending on the complexity of the model to be investigated. There is no known general theory of the solvency of all partial differential equations. Such a theory is implausible, given the large variety of possible models. This issue has led to the development of research on methods of solving partial differential equations problems.
Some cases of partial differential equations that are interesting and continue to be developed in various studies are the Laplace equation, Helmholtz equation, Liouville equation, wave equation, Korteweg-de Vries equation [49] [46]. As for the system of partial differential equations, especially in the case of fluid mechanics, the Navier-Stokes system of nonlinear equations and their linearized form, namely the Neumann-Kelvin system of equations, is still being developed to model the dynamics of fluid motion. Given the wide variety of cases that can be modelled with partial differential equations of varying degrees of difficulty, the question of what it means to "solve" a partial differential equation may be subtle. Classical and analytical solutions are perfect solutions to the problem of partial differential equations. From these solutions, it can be seen how the dynamics of the model under study are. However, not all problems of differential equations can be obtained with analytical solutions. The informal idea in building an excellent partial differential equation problem model is a model that captures many of the desired features and meets the following three criteria [12] (a) the problem has a solution; (b) the solution is single; (c) the solution depends continuously on the data given in the problem. The last condition is very important for problems arising from physical applications. Meanwhile, observing the existence and singularity of solutions is the main task for mathematicians and classifying and characterizing solutions.
Next is to define what is meant by 'solution'. The desired solution satisfies (a) (c). However, does the solution have to be genuinely analytic or at least infinitely differentiable? This solution is the perfect condition, but it will not be easy to achieve in some cases. Perhaps it would be wiser if the solutions of partial differential equations of order k could be differentiated as much as k times continuously. Then at least all the derivatives that appear in the problem of partial differential equations will be continuous, although it is possible that certain higher derivatives will not exist. The idea of solving a partial differential equation problem has always been an exciting topic to study. In this study, variational methods will be studied to analyze the existence and singularity of a solution qualitatively.
The principle of the variational approach to solving partial differential equations is to replace the equation with an equivalent formula called variational, which is obtained by integrating the equation multiplied by any function, called the test function. The solution generated by this method is a weak solution of partial differential equations. Through the Lax-Milgram Theorem, it can be shown that the solution exists and is singular. Then through the regulatory process, it will be proven that the weak solution is a strong solution to the problem of partial differential equations. This variational method is very useful in determining the existence and singularity of solutions to the boundary condition problems of partial differential equations. The applications of this method can be found in the article [49], [45]. In that article, they analyzed weak solutions by functional analysis to prove the existence of a single solution of the linear boundary value problem that describes the motion of the equilibrium state of a half-submerged cylinder in an ideal, incompressible heavy fluid. We can also refer to the following articles: [53], [62], [17], [56], [44], [57], [16] to better understand the concept of variational method.
This study provides an alternative method in analyzing the problem model of partial differential equations analytically. The partial differential equation modelling that will be analyzed in this research is built from fluid dynamics modelling.

Finite element method for one dimensional case
We seek to calculate an approximation of the solution [57] u : [0, 1] → R of the following problem: with > 0, λ and f are given so that there is a unique solution to this problem and we want to approach this solution by a continuous function, polynomial by parts.
(1) Write the variational formulation of the problem We construct the variational formulation by multiplying both side of (1) by a test function v satisfying the boundary conditions and then integrate over [0, 1].
We thus obtain Integrating by parts, and since v(0)=v(1)=0 then we have We obtain the bilinear and linear form: Therefore we can express the variational formulation as follows: find u ∈

) Calculation of the exact solution
We suppose the function f is constant not null.
(a) Find the exact solution of the problem.
-Solve for r: -Solution : -Subtitute the boundary conditions, we obtain the exact solution of (1) and (2): Let us generate a mesh, let a uniform Cartesian mesh x i = ih, i = 0, 1, 2, · · · , n where h = 1/n, defining the intervals (x i−1 , x i ), i = 1, 2, · · · , n. Then, we construct a set of basis functions bassed on the mesh, such as the piecewise linear functions i = 1, 2, · · · , n − 1 Represent the approximate (FE) solution by a linear combination of the basis functions where the coefficients c j are the unknowns to be determined. Substituting the approximate solution u h (x) in the variational formulation: Let us define a finite dimensional space on the mesh. Let the solution be in the space V , which is H 1 0 (0, 1) in the model problem. Based on the mesh, (b) Show that the approximate solution u h can be found as the solution of a linear system where B h is a symmetric tridiagonal matrix and C h is an antisymmetric tridiagonal matrix. From (2.8), we have We choose the test function v(x) as φ 1 , φ 2 , · · · , φ n−1 successively, in the matrix vector form: Existence and uniqueness in the linearised problems of PDP 147 Then we can write the approximation solution u h can be obtained by solve the following linear system: By standard calculation of integral, we will get B h is a symmetric tridiagonal matrix and C h is an antisymmetric tridiagonal matrix. Let us see the case: = 0.1, λ = 1, f = 1, n = 10:

Then we have
We also have

Hence by the calculation of
(2.12) Existence and uniqueness in the linearised problems of PDP 149 For n = 20, we can obtain the solution with the similar calculation as follows: Study of the error. Fix = 0.1, λ = 1, f = 1. For n ranging from 10 to 100 (in steps of 10), plot the curves log n → log ||e n || ∞ where e n is the error vector defined by e n (k) = u h (k) − u(x k ). Deduce a decay law of ||e n || ∞ of the form ||e n || ∞ ∼ Constant/n s1 where s 1 is to be determined.

We seek an approximate solution (finite element method P2)
The basis function for P2 is quadratic function. Let us generate a mesh, let a uniform Cartesian mesh x i = ih, i = 0, 1, 2, · · · , n where h = 1/n, defining the intervals (x i−1 , x i ), i = 1, 2, · · · , n. Then, we construct two sets of functions bassed on the mesh, such as the quadratic functions i = 1, 2, · · · , n − 1. The first function is: . (2.14) The second function is for fill the middle value of the first function for j = 0, 1, 2, ..., n − 1 Then we construct the basis function as the combination of that two functions such as: We can plot the basis function as follows (for n=10): where the coefficients c j are the unknowns to be determined. For the case, n = 4, = 0.1, λ = 1, f = 1 we obtain these results: Then we have Now let us check for n = 10: Existence and uniqueness in the linearised problems of PDP 153  Figure 9. Comparison result between exact solution and approximation solution for P 2(n = 10)

Variational method for two dimensional case
Let Ω an open bounded related of R N , N ≥ 1. For v ∈ L 2 (Ω), we denotē for g ∈ L 2 (Ω).
Proof. We will apply the Lax-Milgram theorem [24] by checking the continuity of bilinear and linear form and also coercivity of the bilinear form. From (3.4) we have the bilinear form is The coercivity is satistied by the norm of V .
Finally we obtain the continuity of L by Cauchy-Schwarz inequality, (3.14) According to the Lax-Milgram theorem, we obtain that there exists a unique function u ∈ V such that for all v ∈ V , we have B[u, v] = L(v).
By using the FreeFem software, we can simulate the solution as follows:

Acknowledgements
Financial support from the DIPA Fakultas MIPA Universitas Tanjungpura 2021 is gratefully acknowledged.