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100-603: Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important in applied mathematics , where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations . For example, in quantum mechanics ,

200-541: A b y ( x ) z ( x ) w ( x ) d x , {\displaystyle {\bigl \langle }y(x),z(x){\bigr \rangle }=\int _{a}^{b}y(x)z(x)w(x)\,dx,} w ( x ) = exp ⁡ ( ∫ g ( x ) f ( x ) d x ) f ( x ) . {\displaystyle w(x)={\frac {\exp \left(\int {\frac {g(x)}{f(x)}}\,dx\right)}{f(x)}}.} Applied mathematics Applied mathematics

300-505: A n = ⟨ X n ( x ) , s ( x ) ⟩ ⟨ X n ( x ) , X n ( x ) ⟩ {\displaystyle a_{n}={\frac {{\bigl \langle }X_{n}(x),s(x){\bigr \rangle }}{{\bigl \langle }X_{n}(x),X_{n}(x){\bigr \rangle }}}} where ⟨ y ( x ) , z ( x ) ⟩ = ∫

400-682: A n exp ⁡ ( λ n t − ∫ 0 t k ( τ ) d τ ) {\displaystyle T_{n}(t)=a_{n}\exp \left(\lambda _{n}t-\int _{0}^{t}k(\tau )\,d\tau \right)} u ( x , t ) = ∑ n a n X n ( x ) exp ⁡ ( λ n t − ∫ 0 t k ( τ ) d τ ) {\displaystyle u(x,t)=\sum _{n}a_{n}X_{n}(x)\exp \left(\lambda _{n}t-\int _{0}^{t}k(\tau )\,d\tau \right)}

500-466: A , b ∈ R {\displaystyle a,b\in \mathbb {R} } ) and x {\displaystyle x} denotes the Cartesian product , square brackets denote closed intervals , then there is an interval I = [ x 0 − h , x 0 + h ] ⊂ [ x 0 − a , x 0 +

600-1291: A , t ) = u ( b , t ) = 0 , u ( x , 0 ) = s ( x ) . {\displaystyle u(a,t)=u(b,t)=0,\qquad u(x,0)=s(x).} Separating variables, we assume that u ( x , t ) = X ( x ) T ( t ) . {\displaystyle u(x,t)=X(x)T(t).} Then our above partial differential equation may be written as: L ^ X ( x ) X ( x ) = M ^ T ( t ) T ( t ) {\displaystyle {\frac {{\hat {L}}X(x)}{X(x)}}={\frac {{\hat {M}}T(t)}{T(t)}}} where L ^ = f ( x ) d 2 d x 2 + g ( x ) d d x + h ( x ) , M ^ = d d t + k ( t ) . {\displaystyle {\hat {L}}=f(x){\frac {d^{2}}{dx^{2}}}+g(x){\frac {d}{dx}}+h(x),\qquad {\hat {M}}={\frac {d}{dt}}+k(t).} Since, by definition, L̂ and X ( x ) are independent of time t and M̂ and T ( t ) are independent of position x , then both sides of

700-445: A ] {\displaystyle I=[x_{0}-h,x_{0}+h]\subset [x_{0}-a,x_{0}+a]} for some h ∈ R {\displaystyle h\in \mathbb {R} } where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F {\displaystyle F} to be linear, this applies to non-linear equations that take

800-454: A linear operator L mapping a function u to another function Lu , and it can be studied in the context of functional analysis . In fact, equation ( 1 ) can be written as L u = λ u . {\displaystyle Lu=\lambda u.} This is precisely the eigenvalue problem; that is, one seeks eigenvalues λ 1 , λ 2 , λ 3 ,... and the corresponding eigenvectors u 1 , u 2 , u 3 ,... of

900-526: A Sturm–Liouville problem in terms of the eigenfunctions X n ( x ) and eigenvalues λ n . The second of these equations can be analytically solved once the eigenvalues are known. d d t T n ( t ) = ( λ n − k ( t ) ) T n ( t ) {\displaystyle {\frac {d}{dt}}T_{n}(t)={\bigl (}\lambda _{n}-k(t){\bigr )}T_{n}(t)} T n ( t ) =

1000-924: A basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies ω mn . This representation may require a convergent infinite sum. Consider a linear second-order differential equation in one spatial dimension and first-order in time of the form: f ( x ) ∂ 2 u ∂ x 2 + g ( x ) ∂ u ∂ x + h ( x ) u = ∂ u ∂ t + k ( t ) u , {\displaystyle f(x){\frac {\partial ^{2}u}{\partial x^{2}}}+g(x){\frac {\partial u}{\partial x}}+h(x)u={\frac {\partial u}{\partial t}}+k(t)u,} u (

1100-489: A closed rectangle R = [ x 0 − a , x 0 + a ] × [ y 0 − b , y 0 + b ] {\displaystyle R=[x_{0}-a,x_{0}+a]\times [y_{0}-b,y_{0}+b]} in the x − y {\displaystyle x-y} plane, where a {\displaystyle a} and b {\displaystyle b} are real (symbolically:

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1200-1326: A constant  λ . The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0 , L 1 or y = 0 , L 2 and define the simplest possible Sturm–Liouville eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence, W m n ( x , y , t ) = A m n sin ⁡ ( m π x L 1 ) sin ⁡ ( n π y L 2 ) cos ⁡ ( ω m n t ) {\displaystyle W_{mn}(x,y,t)=A_{mn}\sin \left({\frac {m\pi x}{L_{1}}}\right)\sin \left({\frac {n\pi y}{L_{2}}}\right)\cos \left(\omega _{mn}t\right)} where m and n are non-zero integers , A mn are arbitrary constants, and ω m n 2 = c 2 ( m 2 π 2 L 1 2 + n 2 π 2 L 2 2 ) . {\displaystyle \omega _{mn}^{2}=c^{2}\left({\frac {m^{2}\pi ^{2}}{L_{1}^{2}}}+{\frac {n^{2}\pi ^{2}}{L_{2}^{2}}}\right).} The functions W mn form

1300-479: A department of mathematical sciences (particularly at colleges and small universities). Actuarial science applies probability, statistics, and economic theory to assess risk in insurance, finance and other industries and professions. Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. The applied methods usually refer to nontrivial mathematical techniques or approaches. Mathematical economics

1400-710: A differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics . Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modeling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modeling (population competition), economics (stock trends, interest rates and

1500-425: A novel approach, subsequently elaborated by Thomé and Frobenius . Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those in his theory of Abelian integrals . As the latter can be classified according to the properties of the fundamental curve that remains unchanged under

1600-638: A periodic function, has a jump at  π ) converges to the average of the left and right limits (see convergence of Fourier series ). Therefore, by using formula ( 4 ), we obtain the solution: y = ∑ k = 1 ∞ 2 ( − 1 ) k k 3 sin ⁡ k x = 1 6 ( x 3 − π 2 x ) . {\displaystyle y=\sum _{k=1}^{\infty }2{\frac {(-1)^{k}}{k^{3}}}\sin kx={\tfrac {1}{6}}(x^{3}-\pi ^{2}x).} In this case, we could have found

1700-426: A rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces f = 0 {\displaystyle f=0} under rational one-to-one transformations. From 1870, Sophus Lie 's work put the theory of differential equations on a better foundation. He showed that the integration theories of

1800-479: A solution to the proposed equation is evidently: y = ∑ i α i λ i u i . {\displaystyle y=\sum _{i}{\frac {\alpha _{i}}{\lambda _{i}}}u_{i}.} This solution will be valid only over the open interval a < x < b , and may fail at the boundaries. Consider the Sturm–Liouville problem: for

1900-516: A union of "new" mathematical applications with the traditional fields of applied mathematics. With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable. Historically, mathematics was most important in the natural sciences and engineering . However, since World War II , fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory , which grew out of economic considerations. Further,

2000-513: A unique maximum (possibly infinite) open interval such that any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain I max {\displaystyle I_{\max }} . In the case that x ± ≠ ± ∞ {\displaystyle x_{\pm }\neq \pm \infty } , there are exactly two possibilities where Ω {\displaystyle \Omega }

2100-452: Is "regular". The problem is said to be regular if: The function w = w ( x ) {\displaystyle w=w(x)} , sometimes denoted r = r ( x ) {\displaystyle r=r(x)} , is called the weight or density function. The goals of a Sturm–Liouville problem are: For a regular Sturm–Liouville problem, a function y = y ( x ) {\displaystyle y=y(x)}

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2200-452: Is a differential equation (DE) dependent on only a single independent variable . As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable, and, less commonly, in contrast with stochastic differential equations (SDEs) where

2300-541: Is a vector-valued function of y {\displaystyle \mathbf {y} } and its derivatives, then is an explicit system of ordinary differential equations of order n > {\displaystyle n>} and dimension m {\displaystyle m} . In column vector form: These are not necessarily linear. The implicit analogue is: where 0 = ( 0 , 0 , … , 0 ) {\displaystyle {\boldsymbol {0}}=(0,0,\ldots ,0)}

2400-399: Is a solution containing n {\displaystyle n} arbitrary independent constants of integration . A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set ' initial conditions or boundary conditions '. A singular solution is a solution that cannot be obtained by assigning definite values to

2500-462: Is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined via second-order homogeneous linear equations . The problems are identified as Sturm–Liouville problems (SLP) and are named after J. C. F. Sturm and J. Liouville , who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and

2600-418: Is also called "industrial mathematics". The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics , computational science , and computational engineering , which use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary. Sometimes,

2700-623: Is an interval, is called a solution or integral curve for F {\displaystyle F} , if u {\displaystyle u} is n {\displaystyle n} -times differentiable on I {\displaystyle I} , and Given two solutions u : J ⊂ R → R {\displaystyle u:J\subset \mathbb {R} \to \mathbb {R} } and v : I ⊂ R → R {\displaystyle v:I\subset \mathbb {R} \to \mathbb {R} } , u {\displaystyle u}

2800-641: Is associated with the following mathematical sciences: With applications of applied geometry together with applied chemistry. Scientific computing includes applied mathematics (especially numerical analysis ), computing science (especially high-performance computing ), and mathematical modelling in a scientific discipline. Computer science relies on logic , algebra , discrete mathematics such as graph theory , and combinatorics . Operations research and management science are often taught in faculties of engineering, business, and public policy. Applied mathematics has substantial overlap with

2900-679: Is based on statistics, probability, mathematical programming (as well as other computational methods ), operations research, game theory, and some methods from mathematical analysis. In this regard, it resembles (but is distinct from) financial mathematics , another part of applied mathematics. According to the Mathematics Subject Classification (MSC), mathematical economics falls into the Applied mathematics/other classification of category 91: with MSC2010 classifications for ' Game theory ' at codes 91Axx Archived 2015-04-02 at

3000-404: Is called a solution if it is continuously differentiable and satisfies the equation ( 1 ) at every x ∈ ( a , b ) {\displaystyle x\in (a,b)} . In the case of more general p , q , w {\displaystyle p,q,w} , the solutions must be understood in a weak sense . The terms eigenvalue and eigenvector are used because

3100-433: Is called an extension of v {\displaystyle v} if I ⊂ J {\displaystyle I\subset J} and A solution that has no extension is called a maximal solution . A solution defined on all of R {\displaystyle \mathbb {R} } is called a global solution . A general solution of an n {\displaystyle n} th-order equation

Sturm–Liouville theory - Misplaced Pages Continue

3200-435: Is defined on sufficiently smooth functions which satisfy the above regular boundary conditions. Moreover, L is a self-adjoint operator: ⟨ L f , g ⟩ = ⟨ f , L g ⟩ . {\displaystyle \langle Lf,g\rangle =\langle f,Lg\rangle .} This can be seen formally by using integration by parts twice, where the boundary terms vanish by virtue of

3300-419: Is frequently used when discussing the method of undetermined coefficients and variation of parameters . For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on

3400-463: Is left to solve: L y = f . {\displaystyle Ly=f.} In general, if initial conditions at some point are specified, for example y ( a ) = 0 and y ′( a ) = 0 , a second order differential equation can be solved using ordinary methods and the Picard–Lindelöf theorem ensures that the differential equation has a unique solution in a neighbourhood of the point where

3500-525: Is more useful for differentiation and integration , whereas Lagrange's notation y ′ , y ″ , … , y ( n ) {\displaystyle y',y'',\ldots ,y^{(n)}} is more useful for representing higher-order derivatives compactly, and Newton's notation ( y ˙ , y ¨ , y . . . ) {\displaystyle ({\dot {y}},{\ddot {y}},{\overset {...}{y}})}

3600-613: Is not an eigenvalue. Then, computing the resolvent amounts to solving a nonhomogeneous equation, which can be done using the variation of parameters formula. This shows that the resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem). As a consequence of the Arzelà–Ascoli theorem , this integral operator is compact and existence of a sequence of eigenvalues α n which converge to 0 and eigenfunctions which form an orthonormal basis follows from

3700-587: Is often used in physics for representing derivatives of low order with respect to time. Given F {\displaystyle F} , a function of x {\displaystyle x} , y {\displaystyle y} , and derivatives of y {\displaystyle y} . Then an equation of the form is called an explicit ordinary differential equation of order n {\displaystyle n} . More generally, an implicit ordinary differential equation of order n {\displaystyle n} takes

3800-525: Is probably the most widespread mathematical science used in the social sciences . Academic institutions are not consistent in the way they group and label courses, programs, and degrees in applied mathematics. At some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and (Pure) Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, but many undergraduate-only institutions include statistics under

3900-810: Is said to be in Sturm–Liouville form or self-adjoint form . All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of ( 1 ) by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations , or if y is a vector ). Some examples are below. x 2 y ″ + x y ′ + ( x 2 − ν 2 ) y = 0 {\displaystyle x^{2}y''+xy'+\left(x^{2}-\nu ^{2}\right)y=0} which can be written in Sturm–Liouville form (first by dividing through by x , then by collapsing

4000-785: Is the zero vector . In matrix form For a system of the form F ( x , y , y ′ ) = 0 {\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}} , some sources also require that the Jacobian matrix ∂ F ( x , u , v ) ∂ v {\displaystyle {\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}} be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In

4100-458: Is the application of mathematical methods by different fields such as physics , engineering , medicine , biology , finance , business , computer science , and industry . Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models . In

Sturm–Liouville theory - Misplaced Pages Continue

4200-517: Is the open set in which F {\displaystyle F} is defined, and ∂ Ω ¯ {\displaystyle \partial {\bar {\Omega }}} is its boundary. Note that the maximum domain of the solution This means that F ( x , y ) = y 2 {\displaystyle F(x,y)=y^{2}} , which is C 1 {\displaystyle C^{1}} and therefore locally Lipschitz continuous, satisfying

4300-567: Is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are " square-summable ", the Fourier series converges in L which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier series converge at every point of differentiability, and at jump points (the function x , considered as

4400-554: The L operator. The proper setting for this problem is the Hilbert space L 2 ( [ a , b ] , w ( x ) d x ) {\displaystyle L^{2}([a,b],w(x)\,dx)} with scalar product ⟨ f , g ⟩ = ∫ a b f ( x ) ¯ g ( x ) w ( x ) d x . {\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)w(x)\,dx.} In this space L

4500-759: The Lucasian Professor of Mathematics whose past holders include Isaac Newton , Charles Babbage , James Lighthill , Paul Dirac , and Stephen Hawking . Schools with separate applied mathematics departments range from Brown University , which has a large Division of Applied Mathematics that offers degrees through the doctorate , to Santa Clara University , which offers only the M.S. in applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT . Students in this program also learn another skill (computer science, engineering, physics, pure math, etc.) to supplement their applied math skills. Applied mathematics

4600-675: The Wayback Machine and for 'Mathematical economics' at codes 91Bxx Archived 2015-04-02 at the Wayback Machine . The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business , engineering , physics , chemistry , psychology , biology , computer science , scientific computation , information theory , and mathematical physics . Ordinary differential equation In mathematics , an ordinary differential equation ( ODE )

4700-534: The spectral theorem for compact operators . Finally, note that ( L − z ) − 1 u = α u , L u = ( z + α − 1 ) u , {\displaystyle \left(L-z\right)^{-1}u=\alpha u,\qquad Lu=\left(z+\alpha ^{-1}\right)u,} are equivalent, so we may take λ = z + α − 1 {\displaystyle \lambda =z+\alpha ^{-1}} with

4800-617: The "applications of mathematics" within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however, mathematical biologists have posed problems that have stimulated the growth of pure mathematics. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems

4900-469: The Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders. The behavior of a system of ODEs can be visualized through the use of a phase portrait . Given a differential equation a function u : I ⊂ R → R {\displaystyle u:I\subset \mathbb {R} \to \mathbb {R} } , where I {\displaystyle I}

5000-1798: The Legendre equation is equivalent to ( ( 1 − x 2 ) y ′ ) ′ + ν ( ν + 1 ) y = 0 {\displaystyle \left(\left(1-x^{2}\right)y'\right)'+\nu (\nu +1)y=0} x 3 y ″ − x y ′ + 2 y = 0 {\displaystyle x^{3}y''-xy'+2y=0} Divide throughout by x : y ″ − 1 x 2 y ′ + 2 x 3 y = 0 {\displaystyle y''-{\frac {1}{x^{2}}}y'+{\frac {2}{x^{3}}}y=0} Multiplying throughout by an integrating factor of μ ( x ) = exp ⁡ ( ∫ − d x x 2 ) = e 1 / x , {\displaystyle \mu (x)=\exp \left(\int -{\frac {dx}{x^{2}}}\right)=e^{{1}/{x}},} gives e 1 / x y ″ − e 1 / x x 2 y ′ + 2 e 1 / x x 3 y = 0 {\displaystyle e^{{1}/{x}}y''-{\frac {e^{{1}/{x}}}{x^{2}}}y'+{\frac {2e^{{1}/{x}}}{x^{3}}}y=0} which can be easily put into Sturm–Liouville form since d d x e 1 / x = − e 1 / x x 2 {\displaystyle {\frac {d}{dx}}e^{{1}/{x}}=-{\frac {e^{{1}/{x}}}{x^{2}}}} so

5100-454: The Sturm–Liouville form L y = f {\displaystyle Ly=f} : writing a general Sturm–Liouville operator as: L u = p w ( x ) u ″ + p ′ w ( x ) u ′ + q w ( x ) u , {\displaystyle Lu={\frac {p}{w(x)}}u''+{\frac {p'}{w(x)}}u'+{\frac {q}{w(x)}}u,} one solves

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5200-467: The above equation must be equal to a constant: L ^ X ( x ) = λ X ( x ) , X ( a ) = X ( b ) = 0 , M ^ T ( t ) = λ T ( t ) . {\displaystyle {\hat {L}}X(x)=\lambda X(x),\qquad X(a)=X(b)=0,\qquad {\hat {M}}T(t)=\lambda T(t).} The first of these equations must be solved as

5300-445: The advancement of science and technology. With the advent of modern times, the application of mathematics in fields such as science, economics, technology, and more became deeper and more timely. The development of computers and other technologies enabled a more detailed study and application of mathematical concepts in various fields. Today, Applied Mathematics continues to be crucial for societal and technological advancement. It guides

5400-406: The answer using antidifferentiation , but this is no longer useful in most cases when the differential equation is in many variables. Certain partial differential equations can be solved with the help of Sturm–Liouville theory. Suppose we are interested in the vibrational modes of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L 1 , 0 ≤ y ≤ L 2 . The equation of motion for

5500-495: The arbitrary constants in the general solution. In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the homogeneous solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the guessing method section in this article, and

5600-456: The author and upon which notation is most useful for the task at hand. In this context, the Leibniz's notation d y d x , d 2 y d x 2 , … , d n y d x n {\displaystyle {\frac {dy}{dx}},{\frac {d^{2}y}{dx^{2}}},\ldots ,{\frac {d^{n}y}{dx^{n}}}}

5700-558: The boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem, one looks at the resolvent ( L − z ) − 1 , z ∈ R , {\displaystyle \left(L-z\right)^{-1},\qquad z\in \mathbb {R} ,} where z

5800-519: The corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations. There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are In their basic form both of these theorems only guarantee local results, though

5900-692: The development of Newtonian physics , and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a pedagogical legacy in the United States: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments. Engineering and computer science departments have traditionally made use of applied mathematics. As time passed, Applied Mathematics grew alongside

6000-613: The development of new technologies, economic progress, and addresses challenges in various scientific fields and industries. The history of Applied Mathematics continually demonstrates the importance of mathematics in human progress. Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography ), though they are not generally considered to be part of

6100-549: The differential equation is equivalent to ( e 1 / x y ′ ) ′ + 2 e 1 / x x 3 y = 0. {\displaystyle \left(e^{{1}/{x}}y'\right)'+{\frac {2e^{{1}/{x}}}{x^{3}}}y=0.} P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 {\displaystyle P(x)y''+Q(x)y'+R(x)y=0} Multiplying through by

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6200-480: The differential equation, and is indicated in the notation F ( x ( t ) ) {\displaystyle F(x(t))} . In what follows, y {\displaystyle y} is a dependent variable representing an unknown function y = f ( x ) {\displaystyle y=f(x)} of the independent variable x {\displaystyle x} . The notation for differentiation varies depending upon

6300-465: The discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory , and makes extensive use of scientific computing, analysis, and optimization ; for the design of experiments , statisticians use algebra and combinatorial design . Applied mathematicians and statisticians often work in

6400-517: The distinction between "application of mathematics" and "applied mathematics". Some universities in the U.K . host departments of Applied Mathematics and Theoretical Physics , but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge , housing

6500-720: The equation into an equivalent linear ODE (see, for example Riccati equation ). Some ODEs can be solved explicitly in terms of known functions and integrals . When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution. Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences . Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that

6600-411: The field of applied mathematics per se . There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and

6700-446: The field of applied mathematics per se . Such descriptions can lead to applicable mathematics being seen as a collection of mathematical methods such as real analysis , linear algebra , mathematical modelling , optimisation , combinatorics , probability and statistics , which are useful in areas outside traditional mathematics and not specific to mathematical physics . Other authors prefer describing applicable mathematics as

6800-545: The first order as accepted circa 1900. The primitive attempt in dealing with differential equations had in view a reduction to quadratures . As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the n {\displaystyle n} th degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that complex differential equations require complex numbers . Hence, analysts began to substitute

6900-684: The first two terms on the left into one term) as ( x y ′ ) ′ + ( x − ν 2 x ) y = 0. {\displaystyle \left(xy'\right)'+\left(x-{\frac {\nu ^{2}}{x}}\right)y=0.} ( 1 − x 2 ) y ″ − 2 x y ′ + ν ( ν + 1 ) y = 0 {\displaystyle \left(1-x^{2}\right)y''-2xy'+\nu (\nu +1)y=0} which can easily be put into Sturm–Liouville form, since ⁠ d / dx ⁠ (1 − x ) = −2 x , so

7000-537: The force F {\displaystyle F} , is given by the differential equation which constrains the motion of a particle of constant mass m {\displaystyle m} . In general, F {\displaystyle F} is a function of the position x ( t ) {\displaystyle x(t)} of the particle at time t {\displaystyle t} . The unknown function x ( t ) {\displaystyle x(t)} appears on both sides of

7100-498: The form F ( x , y ) {\displaystyle F(x,y)} , and it can also be applied to systems of equations. When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely: For each initial condition ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} there exists

7200-542: The form: There are further classifications: A number of coupled differential equations form a system of equations. If y {\displaystyle \mathbf {y} } is a vector whose elements are functions; y ( x ) = [ y 1 ( x ) , y 2 ( x ) , … , y m ( x ) ] {\displaystyle \mathbf {y} (x)=[y_{1}(x),y_{2}(x),\ldots ,y_{m}(x)]} , and F {\displaystyle \mathbf {F} }

7300-449: The initial conditions have been specified. But if in place of specifying initial values at a single point , it is desired to specify values at two different points (so-called boundary values), e.g. y ( a ) = 0 and y ( b ) = 1 , the problem turns out to be much more difficult. Notice that by adding a suitable known differentiable function to y , whose values at a and b satisfy the desired boundary conditions, and injecting inside

7400-1636: The integrating factor μ ( x ) = 1 P ( x ) exp ⁡ ( ∫ Q ( x ) P ( x ) d x ) , {\displaystyle \mu (x)={\frac {1}{P(x)}}\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right),} and then collecting gives the Sturm–Liouville form: d d x ( μ ( x ) P ( x ) y ′ ) + μ ( x ) R ( x ) y = 0 , {\displaystyle {\frac {d}{dx}}\left(\mu (x)P(x)y'\right)+\mu (x)R(x)y=0,} or, explicitly: d d x ( exp ⁡ ( ∫ Q ( x ) P ( x ) d x ) y ′ ) + R ( x ) P ( x ) exp ⁡ ( ∫ Q ( x ) P ( x ) d x ) y = 0. {\displaystyle {\frac {d}{dx}}\left(\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right)y'\right)+{\frac {R(x)}{P(x)}}\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right)y=0.} The mapping defined by: L u = − 1 w ( x ) ( d d x [ p ( x ) d u d x ] + q ( x ) u ) {\displaystyle Lu=-{\frac {1}{w(x)}}\left({\frac {d}{dx}}\left[p(x)\,{\frac {du}{dx}}\right]+q(x)u\right)} can be viewed as

7500-809: The latter can be extended to give a global result, for example, if the conditions of Grönwall's inequality are met. Also, uniqueness theorems like the Lipschitz one above do not apply to DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone. The theorem can be stated simply as follows. For the equation and initial value problem: y ′ = F ( x , y ) , y 0 = y ( x 0 ) {\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})} if F {\displaystyle F} and ∂ F / ∂ y {\displaystyle \partial F/\partial y} are continuous in

7600-509: The market equilibrium price changes). Many mathematicians have studied differential equations and contributed to the field, including Newton , Leibniz , the Bernoulli family , Riccati , Clairaut , d'Alembert , and Euler . A simple example is Newton's second law of motion—the relationship between the displacement x {\displaystyle x} and the time t {\displaystyle t} of an object under

7700-422: The mathematics department. Many applied mathematics programs (as opposed to departments) consist primarily of cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside mathematics, while others require substantial coursework in a specific area of application. In some respects this difference reflects

7800-419: The middle of the nineteenth century has it received special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably Casorati and Cayley . To the latter is due (1872) the theory of singular solutions of differential equations of

7900-623: The older mathematicians can, using Lie groups , be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable integration difficulties. He also emphasized the subject of transformations of contact . Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations. A general solution approach uses

8000-414: The one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem. Sturm–Liouville theory is named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), who developed the theory. The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem on a finite interval [ a , b ] {\displaystyle [a,b]} that

8100-601: The one-dimensional time-independent Schrödinger equation is a special case of a Sturm–Liouville equation. Consider a general inhomogeneous second-order linear differential equation P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = f ( x ) {\displaystyle P(x)y''+Q(x)y'+R(x)y=f(x)} for given functions P ( x ) , Q ( x ) , R ( x ) , f ( x ) {\displaystyle P(x),Q(x),R(x),f(x)} . As before, this can be reduced to

8200-581: The other. Some mathematicians emphasize the term applicable mathematics to separate or delineate the traditional applied areas from new applications arising from fields that were previously seen as pure mathematics. For example, from this viewpoint, an ecologist or geographer using population models and applying known mathematics would not be doing applied, but rather applicable, mathematics. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography ), though they are not generally considered to be part of

8300-646: The past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis , most notably differential equations ; approximation theory (broadly construed, to include representations , asymptotic methods, variational methods , and numerical analysis ); and applied probability . These areas of mathematics related directly to

8400-649: The progression is random. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a 0 ( x ) , … , a n ( x ) {\displaystyle a_{0}(x),\ldots ,a_{n}(x)} and b ( x ) {\displaystyle b(x)} are arbitrary differentiable functions that do not need to be linear, and y ′ , … , y ( n ) {\displaystyle y',\ldots ,y^{(n)}} are

8500-693: The proposed differential equation, it can be assumed without loss of generality that the boundary conditions are of the form y ( a ) = 0 and y ( b ) = 0 . Here, the Sturm–Liouville theory comes in play: indeed, a large class of functions f can be expanded in terms of a series of orthonormal eigenfunctions u i of the associated Liouville operator with corresponding eigenvalues λ i : f ( x ) = ∑ i α i u i ( x ) , α i ∈ R . {\displaystyle f(x)=\sum _{i}\alpha _{i}u_{i}(x),\quad \alpha _{i}\in {\mathbb {R} }.} Then

8600-662: The same boundary conditions y ( 0 ) = y ( π ) = 0 {\displaystyle y(0)=y(\pi )=0} . In this case, we must expand f ( x ) = x as a Fourier series. The reader may check, either by integrating ∫ e x dx or by consulting a table of Fourier transforms, that we thus obtain L y = ∑ k = 1 ∞ − 2 ( − 1 ) k k sin ⁡ k x . {\displaystyle Ly=\sum _{k=1}^{\infty }-2{\frac {\left(-1\right)^{k}}{k}}\sin kx.} This particular Fourier series

8700-408: The same eigenfunctions. If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case, the spectrum no longer consists of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in quantum mechanics , since

8800-609: The same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order , which makes

8900-422: The simple form W = X ( x ) × Y ( y ) × T ( t ) . For such a function W the partial differential equation becomes ⁠ X ″ / X ⁠ + ⁠ Y ″ / Y ⁠ = ⁠ 1 / c ⁠ ⁠ T ″ / T ⁠ . Since the three terms of this equation are functions of x , y , t separately, they must be constants. For example, the first term gives X ″ = λX for

9000-541: The solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate Hilbert space of functions with inner product defined using the weight function. Sturm–Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in the function space. The main result of Sturm–Liouville theory states that, for any regular Sturm–Liouville problem: The differential equation ( 1 )

9100-522: The solutions of a Sturm–Liouville problem form an orthogonal basis , and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the Sturm–Liouville problem in this case has no other eigenvectors. Given the preceding, let us now solve the inhomogeneous problem L y = x , x ∈ ( 0 , π ) {\displaystyle Ly=x,\qquad x\in (0,\pi )} with

9200-464: The study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and, if so, what are the characteristic properties. Two memoirs by Fuchs inspired

9300-694: The successive derivatives of the unknown function y {\displaystyle y} of the variable x {\displaystyle x} . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function ). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming

9400-620: The symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions ( Lie theory ). Continuous group theory , Lie algebras , and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs , recursion operators, Bäcklund transform , and finally finding exact analytic solutions to DE. Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines. Sturm–Liouville theory

9500-958: The system: p = P w , p ′ = Q w , q = R w . {\displaystyle p=Pw,\quad p'=Qw,\quad q=Rw.} It suffices to solve the first two equations, which amounts to solving ( Pw )′ = Qw , or w ′ = Q − P ′ P w := α w . {\displaystyle w'={\frac {Q-P'}{P}}w:=\alpha w.} A solution is: w = exp ⁡ ( ∫ α d x ) , p = P exp ⁡ ( ∫ α d x ) , q = R exp ⁡ ( ∫ α d x ) . {\displaystyle w=\exp \left(\int \alpha \,dx\right),\quad p=P\exp \left(\int \alpha \,dx\right),\quad q=R\exp \left(\int \alpha \,dx\right).} Given this transformation, one

9600-480: The term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today, although there is no consensus as to a precise definition. Mathematicians often distinguish between "applied mathematics" on the one hand, and the "applications of mathematics" or "applicable mathematics" both within and outside of science and engineering, on

9700-435: The unknowns are λ and u ( x ) . For boundary conditions, we take for example: u ( 0 ) = u ( π ) = 0. {\displaystyle u(0)=u(\pi )=0.} Observe that if k is any integer, then the function u k ( x ) = sin ⁡ k x {\displaystyle u_{k}(x)=\sin kx} is a solution with eigenvalue λ = k . We know that

9800-531: The utilization and development of mathematical methods expanded into other areas leading to the creation of new fields such as mathematical finance and data science . The advent of the computer has enabled new applications: studying and using the new computer technology itself ( computer science ) to study problems arising in other areas of science (computational science) as well as the mathematics of computation (for example, theoretical computer science , computer algebra , numerical analysis ). Statistics

9900-633: The vertical membrane's displacement, W ( x , y , t ) is given by the wave equation : ∂ 2 W ∂ x 2 + ∂ 2 W ∂ y 2 = 1 c 2 ∂ 2 W ∂ t 2 . {\displaystyle {\frac {\partial ^{2}W}{\partial x^{2}}}+{\frac {\partial ^{2}W}{\partial y^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}W}{\partial t^{2}}}.} The method of separation of variables suggests looking first for solutions of

10000-409: The whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of Lipschitz differential equations. As example, the equation: Admits the finite duration solution: The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since

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